The impulse formula is J = F Δt: impulse equals the average force acting on an object multiplied by the time interval over which that force acts. Because force is the rate of change of momentum, the same impulse also equals the momentum change, J = Δp = mv – mu. Impulse is a vector measured in newton-seconds (N·s).

Drop your phone on a tiled floor and it shatters. Drop it on a thick rug from the same height and it survives. The phone hits the ground with exactly the same momentum both times — so the change in momentum, and therefore the impulse, is identical.

What changes is how long the stop takes. The rug stretches that stop over maybe ten times longer, and the force collapses by the same factor. That single trade-off is what the impulse formula lets you put a number on.

What Is the Impulse Formula?

The impulse formula is J = F Δt, where F is the average force in newtons and Δt is the contact time in seconds. Multiply the two and you get impulse in newton-seconds.

There is a second version of the same equation, and you will use it just as often. Since a force acting for a time is exactly what changes an object’s momentum, impulse can also be worked out from mass and velocity alone: J = mv – mu.

Which version you reach for depends entirely on what the question hands you. Force and a time? Use F Δt.

Mass and two velocities? Use the momentum difference. Both routes give the same number in the same units.

One detail that trips people up in the first line of their working: impulse is a vector. It points in the direction of the force, and in one-dimensional problems that means it carries a sign.

The Impulse Formula and Every Rearrangement You Need

Here are the two forms of the impulse equation, followed by every rearrangement an exam or a lab report will ask for.

Every symbol, with its SI unit:

Rearranging for force or time

• Average force: F = J / Δt
• Contact time: Δt = J / F
• Final velocity: v = u + J / m
• Mass: m = J / (v – u)

Those four rearrangements cover essentially every impulse question you will meet. If you would rather check your algebra against a machine than trust it, our Impulse Calculator solves for impulse, force or time and shows the substitution line by line.

Why N·s and kg·m/s are the same unit

A newton is a kg·m/s², so a newton-second is kg·m/s² × s = kg·m/s. The two units are dimensionally identical, and both describe impulse.

In practice, use N·s when you calculated from force and time, and kg·m/s when you calculated from mass and velocity. Neither is wrong — examiners accept both.

How to Calculate Impulse in 4 Steps

To calculate impulse, pick the form of the formula that matches your data, convert everything to SI units, multiply while keeping track of sign, then quote the answer with a direction. Here is the routine in full.

1. Decide which pair you have. Force plus a time means J = F Δt. Mass plus two velocities means J = mv – mu. If you have all four, you can do it either way and cross-check.
2. Convert to SI and choose a positive direction. Grams to kilograms, milliseconds to seconds, km/h to m/s. Write down which way you are calling positive before any numbers go in.
3. Multiply or subtract, carrying the signs. A rebound means u and v have opposite signs, so the subtraction becomes an addition of magnitudes.
4. Quote units and direction, then sanity-check. An impulse of 4 N·s on a tennis ball is reasonable; 4000 N·s is not. Divide by the mass — if the implied velocity change is absurd, hunt for the slipped decimal point.

Step 4 is the one students skip and the one that catches real errors. A quick division by mass turns an abstract answer back into a velocity you can judge by eye.

Try it live below. Drag the change-in-momentum slider to fix the impulse, then stretch the contact time and watch the average force fall while the shaded area stays exactly the same.

How to Find Impulse from a Force-Time Graph

Impulse is the area under a force-time graph. That one sentence replaces the whole formula, because it works whether the force is constant, spiky or wildly irregular.

When the force is constant, the area is a rectangle and you are back to J = F Δt. When it is not constant, you break the shape into pieces you can measure — and the answer is still in N·s, because the axes are newtons and seconds.

Four common force-time shapes. Whatever the outline, the shaded area under it is the impulse in newton-seconds.

Working with an irregular curve

Real collisions produce a lopsided bump, not a neat triangle. Two methods handle it without calculus.

• Count squares. Work out the impulse of one grid square (its width in seconds × its height in newtons), count the squares under the curve, and multiply. Half-covered squares count as a half.
• Split into strips. Chop the curve into trapezium slices, find each area, and add them. Narrower strips give a closer answer.

If you have already met the area-under-a-graph trick in motion graphs, this is the same skill on different axes — velocity-time gives displacement, force-time gives impulse.

Average Force vs Peak Force: The Number Students Get Wrong

The F in J = F Δt is the average force over the contact, never the peak force. Substituting the peak value is the single most common way an otherwise correct impulse calculation ends up wrong.

Picture a symmetric triangular spike — force climbing from zero to a maximum and back down. Its area is ½ × peak × Δt, so the average force is exactly half the peak.

Use the peak instead and your answer is out by a factor of two. In a crash-safety context, that is the difference between a survivable deceleration and a fatal one.

This is also the reason impulse is the practical way to estimate collision forces at all. As Georgia State University’s HyperPhysics notes on the impulse of force point out, the force during an impact is almost never measurable directly — but the momentum change and the contact time usually are.

Both shaded blocks have the same area, so both deliver the same 980 N·s impulse to a 70 kg driver stopping from 14 m/s. Stretching the stop from 0.02 s to 0.35 s cuts the force by a factor of 17.5.

That is the whole design brief for an airbag, a crash mat and a climbing rope. None of them reduces the impulse — the momentum change is fixed by the crash. They only stretch Δt.

Where the Impulse Formula Gets Used in Practice

The impulse formula turns up wherever a force acts briefly and you need a number for it. These are worked from the assumptions in each row, so you can follow the arithmetic rather than take it on trust.

Rockets: impulse you can buy off a shelf

Rocket engineers care about impulse more than thrust, because thrust alone says nothing about how long it lasts. NASA defines the total impulse of a rocket as the average thrust multiplied by the total firing time — J = F Δt with a rocket motor in place of a tennis racket.

That is why motors are sold by total impulse in newton-seconds. It is the honest measure of how much momentum the motor can hand the vehicle.

Continuous jets: impulse per second

A hose firing 12 kg of water per second at 20 m/s onto a wall delivers 240 kg·m/s of momentum change every second. Divide impulse by time and you have a steady 240 N force.

Same formula, rearranged. This rate form is how fire hoses kick back and how jet engines push aircraft forward.

4 Mistakes That Wreck Impulse Calculations

Most lost marks on impulse questions come from four specific slips, and none of them is about understanding the physics. They are all about the substitution line.

Mistake 1: using the peak force

J = F Δt needs the time-averaged force. Read the peak off a graph and you will typically double your answer, because a symmetric spike averages half its maximum.

Mistake 2: forgetting the sign flip on a rebound

If a ball arrives at 30 m/s and leaves at 40 m/s the other way, the velocity change is 70 m/s, not 10 m/s. Set your positive direction first, write u = -30 m/s and v = +40 m/s, and let the algebra handle it.

Writing the sign convention down before you substitute costs one line and removes the error entirely.

Mistake 3: leaving time in milliseconds

A contact time of 5 ms is 0.005 s. Substitute 5 instead and any force from F = J / Δt lands a thousand times too small, while any impulse from J = F Δt lands a thousand times too big.

Both wrong answers look tidy enough to survive an unchecked script. Convert the units on the line where you write them down, not later.

Mistake 4: confusing net impulse with the contact force

J = Δp gives the impulse of the resultant force. If a question asks for the force from the floor on a bouncing ball, weight acts during the contact too, so the floor’s force is the net impulse divided by Δt plus the weight.

For a brief, violent impact the weight term is often tiny. Ignore it silently and you will still lose the mark for method.

How the Impulse Formula Connects to Momentum, Force and Energy

The impulse formula is Newton’s second law with the time multiplied through. Newton’s law says force is the rate of change of momentum, F = Δp / Δt, and rearranging that single line gives F Δt = Δp — the impulse-momentum theorem.

If you want the concepts behind that step rather than the arithmetic, our guide to momentum and impulse derives the theorem and explains why it matters, and the write-up on Newton’s second law covers the F = ma form you probably met first.

Impulse and collisions

In a collision, the two objects exert equal and opposite forces on each other for exactly the same length of time. Equal force, equal time, so equal and opposite impulses — which is precisely why momentum is conserved.

That holds whether the collision is bouncy or sticky. The distinction between elastic and inelastic collisions is about kinetic energy, not impulse.

Impulse is not energy

Here is a case worth remembering. A ball bounces off a wall with the same speed it arrived at, so its kinetic energy is unchanged — zero energy transfer.

Its impulse, though, is 2mv, the largest it could possibly be. Momentum depends on velocity linearly and carries direction; energy depends on speed squared and does not. They are different quantities and they behave differently.

Worked Problems

Frequently Asked Questions

How to solve physics problems reliably comes down to one repeatable routine: translate the words into data, draw the situation and fix a sign convention, name the governing principle, rearrange the equation in symbols, substitute values in SI units, then check units, magnitude and limiting cases before writing the answer.

You read the question twice. You understand every word in it, you know exactly which chapter it came from, and the page in front of you is still blank.

That blank-page moment is almost never a knowledge problem. It is a process problem — and process is the one part of physics nobody teaches you directly.

What Does It Mean to Solve a Physics Problem?

Solving a physics problem means converting a described situation into a mathematical model, extracting the unknown from that model, and confirming the result is physically sensible. Three jobs, in that order.

Notice what is missing from that description: recall. Most exam boards hand you a formula sheet, and every serious textbook prints one in the back. The equations were never the scarce resource.

The scarce resource is the mapping — the step where a paragraph about a braking cyclist becomes a diagram, a sign convention and one chosen equation. Students who solve quickly are not remembering more than you. They are running a fixed routine that removes decisions.

And routines can be learned in an afternoon. That is the good news buried inside every “I’m just not a physics person” story.

The 6-Step Framework at a Glance

Here is the whole method in one place. Read it once now, then use the detailed walk-through below to watch each step do real work.

1. Translate the words into listed data and one named unknown.
2. Draw the situation and choose which direction is positive.
3. Choose the principle that governs the situation, then take your equation from it.
4. Rearrange to isolate the unknown, in symbols, before any number appears.
5. Substitute in SI units, carrying units through the arithmetic.
6. Check the units, the magnitude and a limiting case, then round sensibly.

Each step earns its place by blocking one specific, predictable mistake:

The six steps run in order, and a failed check sends you back to the diagram — not to a different formula.

How to Solve Physics Problems: The Six Steps in Detail

Each step below is short on purpose. The skill is not in understanding them — it is in refusing to skip one when you are in a hurry.

Step 1 — Translate the Words into Data

Write out every quantity the question gives you, each with its symbol and its unit, then write the unknown on its own line. Do this even when the problem looks trivial.

Physics questions hide data inside ordinary English. Four phrases account for most of it:

• “Starts from rest” — the initial velocity is zero, u = 0.
• “Smooth surface” — friction is zero.
• “Dropped” — the initial vertical velocity is zero.
• “Comes to a stop” — the final velocity is zero, v = 0.

Miss one of those and the problem genuinely does become unsolvable — not because you lack an equation, but because you are one value short and do not know it. The step takes about twenty seconds and rescues a surprising share of the questions students call impossible.

Step 2 — Draw It and Fix Your Signs

Draw the situation and mark a positive direction on the page before you write a single equation. A sketch is not decoration; it is where the physics gets decided.

What you draw depends on the branch. Mechanics wants a free-body diagram showing every force acting on the object — if you are shaky on which forces belong there, our guide to the types of forces in physics works through contact and field forces one at a time. Optics wants a ray diagram with the normal drawn in. Circuits want the circuit redrawn cleanly, loops and junctions labelled.

Then commit to a sign convention. Because velocity, acceleration and force are all vector quantities, their signs mean nothing except relative to the direction you chose — and a marker cannot read your mind unless you write it down.

A common student slip: flipping the sign of gravity halfway through a thrown-ball problem because the ball “is coming down now”. It is not. If up is positive, the acceleration stays negative for the entire flight, including the instant at the top when the velocity is zero.

Velocity changes sign during the flight. Acceleration does not — which is exactly why the sign convention has to be written down first.

Step 3 — Name the Principle, Not the Formula

Before choosing an equation, say which physical principle governs the situation. Equations are consequences of principles, and naming the principle first is what stops you scanning a formula sheet for matching letters.

The question is short: what is conserved here, or what law connects these quantities? A handful of triggers cover most of an introductory course:

• A force acts and something accelerates — Newton’s second law.
• Two objects interact briefly — momentum is conserved.
• Nothing dissipates energy — mechanical energy is conserved.
• Charge flows around a loop — charge and energy conservation give you Kirchhoff’s rules.

This is also why equations are worth learning in families rather than as a flat list — our breakdown of physics formulas by group organises them by the principle each one descends from, which is the order your memory actually wants them in.

One test tells you whether Step 3 was done honestly. Can you say why that equation applies here without pointing at the letters in it? If not, you are formula-hunting.

Step 4 — Rearrange in Symbols Before You Touch a Number

Isolate the unknown algebraically while every quantity is still a letter. Numbers go in once, at the end, and only into a finished expression.

Take the kinematic relation that links speeds to distance without involving time — one of the SUVAT equations:

If the unknown is the distance, rearrange before substituting anything:

• s — displacement, in metres (m)
• u — initial velocity, in metres per second (m/s)
• v — final velocity, in metres per second (m/s)
• a — constant acceleration, in metres per second squared (m/s²)

Three things improve at once. Mistakes become visible, because a wrong symbolic answer looks wrong in a way that a wrong decimal never does. The structure can be sanity-checked before you commit — double the acceleration here and the distance must halve. And when the question changes the numbers, you reuse the expression instead of redoing the problem.

With the rearranged expression in hand, you can confirm the arithmetic using our SUVAT calculator, which solves for any of s, u, v, a or t and prints the working line by line — a way to check Step 4, never a way to skip it.

Rearranging is the step most students name as their weak point, so drill it on its own. The lab below lets you pick a relationship, choose what to solve for, and watch the rearrangement and the unit check happen side by side.

Step 5 — Substitute in SI Units, Carrying the Units

Convert every quantity to SI units before it enters the equation, then write the units alongside the numbers as you substitute. Two habits, one line of working.

The conversions that catch people are always the same handful: km/h into m/s, grams into kilograms, centimetres into metres, minutes into seconds, degrees Celsius into kelvin, and the electrical prefixes — milliamps, kilohms, microfarads. Our guide to SI units in physics covers the conversion rules in full; the underlying system of seven base units is maintained by NIST, which is why every equation you meet assumes them.

Carrying units also converts your calculator into an error detector. Substitute into F = ma with mass in kilograms and acceleration in metres per second squared and the units multiply out to kg·m/s², which is the newton — so if your working produces anything else, the mistake happened before the arithmetic did.

One temperature warning worth internalising: a temperature difference of 80 °C equals a difference of 80 K exactly, so ΔT needs no conversion. An absolute temperature does. Confusing the two is a classic thermodynamics trap.

Step 6 — Check the Answer Three Ways

Run three fast checks before you write the final line: do the units come out right, is the magnitude believable, and does a limiting case behave sensibly? Each takes seconds and each catches a different class of error.

• Units. The units on both sides must match. In v² = u² + 2as, the right-hand side gives m²/s² plus (m/s²)(m) = m²/s² — consistent, so the equation survives the test.
• Magnitude. Compare against something you know. A person cannot run at 90 m/s. A household appliance does not draw 400 A. If your answer is thousands of times off, the error is usually a unit, not the algebra.
• Limiting case. Push a variable to zero or infinity and see whether the formula still makes sense. Set the acceleration to zero in s = (v² – u²) / (2a) and the expression blows up — correctly, because with no acceleration the speed can never change.

Then round. Your answer cannot be more precise than the least precise measurement you were given, so a question quoting 5.0 m and 9.81 m/s² supports two significant figures, not the nine your calculator offers.

How the Framework Changes Between Mechanics, Circuits and Optics

The six steps never change, but Steps 2, 3 and 5 look different in each branch of physics. Knowing what they turn into is most of what “being good at” a topic means.

Read the table by column rather than by row. The pattern that emerges — draw the situation, invoke a conservation law, watch the prefixes — is the framework itself, wearing five different costumes.

Where This Framework Shows Up Outside the Classroom

The same six steps are what professional practice looks like when calculations carry consequences. Four examples, each leaning on a different step.

Collision investigation. An investigator measuring skid marks works backwards to impact speed using the same relation you rearranged above, with the deceleration coming from the road surface. The reconstruction stands up in court because every assumption was written down at Step 1.

Spacecraft navigation. NASA lost the Mars Climate Orbiter in 1999 because one team’s software produced results in pound-force seconds while the spacecraft’s navigation expected newton-seconds. The physics was right and the mission still failed. That is Step 5, at a cost of hundreds of millions of dollars.

Engineering sign-off. Before trusting a simulation, engineers run the order-of-magnitude check from Step 6 by hand. Software will happily return a beam deflection of forty metres and not care; a human comparing that against the length of the beam will.

Clinical imaging. A radiographer setting exposure works from a physical relationship and a unit-consistent calculation, then applies a magnitude check against expected dose. The check exists precisely because a decimal slip has a patient on the other end of it.

Common Misconceptions About Solving Physics Problems

Four beliefs do more damage than any missing equation.

“It’s about memorising formulas”

Memorised formulas are the cheapest part of the skill, and most exams supply them anyway. What is never supplied is the judgement in Step 3 — deciding which principle governs a situation you have not seen before. Spend your revision time there.

“If I can follow the solution, I can solve it”

Following a worked solution is recognition; solving from a blank page is retrieval, and they are different mental operations. Recognition feels like understanding, which is exactly what makes it dangerous. The only honest test is a blank page and a closed book.

“Put the numbers in early to see what happens”

Substituting early buries your reasoning under arithmetic. Once a line reads 0.2553 rather than V/R, you have lost the ability to spot that the structure is wrong — and you have thrown away the reusable result.

“A negative answer means I made a mistake”

Usually it means the opposite. A negative sign generally reports a direction opposite to the one you chose as positive in Step 2 — a deceleration, a force pointing the other way, a charge flowing back. Read the sign as information, then decide whether it is impossible. Negative mass or negative absolute temperature is an error; negative velocity rarely is.

How to Practise So the Framework Becomes Automatic

Practise the framework deliberately for two or three weeks and it stops being a checklist and becomes the way you read a question. A few habits do most of the work.

• Keep an error log tagged by step number. Every mistake belongs to one of the six steps. After twenty problems the pattern is unmistakable, and most students find they are losing marks in one step, not six.
• Sit with a hard problem for ten minutes before opening the solution. The struggle is what builds retrieval; reading the answer early feels productive and teaches almost nothing.
• When you do open it, read one line at a time. Cover the rest, take the hint, close the book, and continue on your own.
• Redo a solved problem two days later from a blank page. If you cannot, you had recognised it rather than learned it.
• Mix topics in a session. Doing twenty momentum problems in a row trains Step 4 and skips Step 3 entirely, because you already know which principle applies.
• Estimate before calculating. Committing to a guess sharpens the magnitude check and costs nothing.

If you want to see the same discipline set out at university level, the MIT 8.01 online textbook devotes its entire second chapter to units, dimensional analysis, problem solving and estimation — before it teaches any mechanics at all. That ordering is not an accident.

Worked Problems

Six problems, rising in difficulty, each solved with the same six steps. Every one ends with a link to the matching calculator so you can vary the numbers and confirm your own working.

Frequently Asked Questions

The Faraday’s law formula states that the EMF induced in a coil equals the number of turns multiplied by the rate at which magnetic flux through it changes: ε = -N(ΔΦ/Δt). EMF is measured in volts, flux in webers and time in seconds. The minus sign gives direction only, not size.

Wave a magnet through a coil of wire and a voltmeter twitches. Wave the same magnet through twice as fast and the needle jumps roughly twice as far — the magnet is no stronger, only quicker. That single observation is why the equation contains a rate of change rather than a field strength.

The trouble starts when you have to put numbers in. Plenty of students who can explain electromagnetic induction perfectly still drop marks by feeding the equation B when it wants ΔB, or by quietly forgetting the coil’s 350 turns. This page is about the arithmetic — what to substitute, how to rearrange, and how to spot a nonsense answer.

What Is the Faraday’s Law Formula?

The Faraday’s law formula is ε = -N(ΔΦ/Δt): the induced EMF equals the number of turns times the rate of change of magnetic flux through the coil. Everything else in this article is that one line, rearranged.

Notice what the equation does not contain. There is no B on its own and no Φ on its own — only a change in flux divided by the time it took. A coil parked in the strongest magnet you own produces exactly zero volts, because nothing is changing.

Almost every calculation therefore has two stages. First work out the magnetic flux, then work out how fast it changed.

Substituting the flux into Faraday’s law gives the working form you will actually use most often. When only the field strength changes and the coil faces the field square-on, cos θ = 1 and the area is fixed:

This is the version that solves the majority of exam questions. The other versions come from letting A change instead of B, or letting θ change instead of either.

Every Symbol in the Faraday’s Law Formula (and Its SI Unit)

Each symbol in the Faraday’s law formula has one SI unit, and mixing them up is the single biggest source of wrong answers. Learn the table below and half the marks look after themselves.

One term deserves its own name. The product NΦ is called the flux linkage, measured in webers (or weber-turns), and Faraday’s law can be read as “EMF equals the rate of change of flux linkage”.

That reading is worth adopting, because it stops N drifting out of your working. If you calculate the flux linkage before and after, the N is baked in from the start.

Every symbol in the Faraday’s law formula, with the SI unit it must be substituted in.

How to Calculate Induced EMF in Four Steps

To calculate induced EMF, find the flux before and after, subtract to get ΔΦ, divide by the time interval, then multiply by the number of turns. The four steps below work for every standard induced-EMF question.

1. Find the flux at the start and at the end. Use Φ = B·A·cos θ for each moment. Convert cm² to m² and mT to T now, before anything else.
2. Subtract to get ΔΦ. Always final minus initial. If the flux fell, ΔΦ is negative — that is information, not an error.
3. Divide by Δt, then multiply by N. This gives ε = -N(ΔΦ/Δt). Only the time during which the flux was actually changing counts.
4. Sanity-check the size and the sign. Quote the magnitude unless the question asks for direction.

That last step is the one most people skip, and it is the cheapest mark on the page. A rough feel for typical magnitudes catches a slipped power of ten instantly:

• A single loop and a hand-waved magnet: millivolts.
• A few hundred turns and a field switched off in a fraction of a second: a few volts to tens of volts.
• A mains generator coil: hundreds of volts.
• Anything above a few kilovolts from a classroom coil means a unit slipped somewhere.

Once you have a number, it is worth checking your working against our Faraday’s Law Calculator, which solves ε = N·A·(ΔB/Δt) and rearranges for the turns, area, field change or time interval so you can see exactly which substitution went astray.

Drag the turns slider and watch the peak EMF scale in exact proportion. Then drag the frequency slider — the field and the coil are unchanged, yet the voltage climbs, which is the rate-of-change idea made visible.

Rearranging the Faraday’s Law Formula

The Faraday’s law formula rearranges into four useful forms, one for each quantity you might be asked to find. The physics never changes; only the subject of the equation does.

The magnitude signs are dropped in the table on purpose. Rearranging is easier when you work with sizes, then restore direction at the end using the physical argument rather than the algebra.

A practical tip: rearrange before you substitute. Putting numbers in first and then trying to unpick them is how sign errors and stray factors of N creep in.

What the Minus Sign Does to Your Answer

The minus sign in Faraday’s law changes the direction of the induced EMF, never its size. If a question asks “find the induced EMF”, the expected answer is almost always the magnitude — 2.8 V, not -2.8 V.

So what is it doing there? It encodes Lenz’s law: the induced EMF pushes current the way that opposes whatever change created it. Drop the minus sign and the equation would let a coil amplify its own flux forever, which would be free energy.

In practice, treat the sign as a separate question. Compute the size from the numbers, then decide the direction from the physics of the situation.

One caution when you carry the result forward. If you feed a negative EMF into a current calculation, the minus survives — so state your positive direction first, or work with magnitudes and note the direction in words. Remember too that an EMF is an energy-per-charge quantity, closely related to potential difference but produced by a changing flux rather than a chemical cell.

Average EMF or Peak EMF? Choosing the Right Version

Use ΔΦ/Δt when you want the average EMF over an interval, and dΦ/dt when you want the EMF at one instant. Choosing the wrong one is a quiet source of lost marks, because both give “an EMF” and neither looks obviously wrong.

The distinction is really about gradients. Flux linkage plotted against time has a slope at every point, and the induced EMF is that slope.

A steeper flux-linkage graph means a larger induced EMF; a horizontal section means no EMF at all.

A rotating coil is the case where the difference really bites. Its flux varies sinusoidally, so the EMF does too, and the peak value is set by the angular frequency ω = 2πf:

The average over a full cycle is zero, because the EMF spends as long negative as positive. That is why rotating coils are quoted by peak or by RMS value, with the RMS equal to the peak divided by sqrt(2), or about 0.707 of the peak.

This sinusoidal output is precisely why generators produce alternating rather than direct current: the sign of the gradient flips twice per revolution, whatever you do.

Common Mistakes When Using the Faraday’s Law Formula

Most wrong answers in Faraday’s law questions come from five specific substitution errors, not from misunderstanding the physics. Each one is worth checking before you commit an answer.

1. Substituting B instead of ΔB

A field of 0.60 T that grew from 0.12 T gives ΔB = 0.48 T, not 0.60 T. The equation only ever cares about the change. If the question quotes a starting value, it is there to be subtracted.

2. Leaving the area in cm²

Squared units bite twice: 1 cm² is 10^-4 m², not 10^-2. A coil of 25 cm² is 2.5 × 10^-3 m². Getting this wrong scales your answer by a factor of 100.

3. Losing the number of turns

Faraday’s law uses flux linkage, NΦ, not flux. With 350 turns, forgetting N makes your answer 350 times too small — and an 8.0 mV answer to a “few volts” question rarely triggers alarm bells on its own.

4. Measuring θ from the wrong line

The angle in Φ = B·A·cos θ is measured between the field and the normal to the coil, the line sticking straight out of its face. A coil lying flat in a vertical field has θ = 0 and maximum flux, not zero.

5. Using the wrong Δt

Δt is the time the flux took to change, not how long the experiment lasted. If a magnet sits still for 5 s and is then yanked out in 0.10 s, the interval you want is 0.10 s.

How the Faraday’s Law Formula Connects to Other Equations

Faraday’s law sits at the centre of a small family of equations that share its variables. Knowing which neighbour to reach for turns a hard question into an easy one.

Magnetic flux density. Before any flux calculation you need B in teslas, which is the quantity the magnetic field equations supply — for a solenoid, for instance, B depends on the current and the turns per metre.

Motional EMF. When a straight conductor of length L slides at speed v across a perpendicular field, the changing area gives ε = B·L·v. It is Faraday’s law with the area doing the changing, not a separate rule.

Ohm’s law. An EMF on its own drives nothing until the circuit is closed. Once it is, the induced current follows from Ohm’s law as I = ε/R, which is how nearly every multi-part induction question ends.

For a fuller treatment of the phenomenon itself — Lenz’s law, transformers and the rest — see the companion guide to electromagnetic induction. The formal derivation appears in MIT OpenCourseWare’s chapter on Faraday’s law (PDF), and the University of Tennessee’s Faraday’s law module works through the flux-linkage sign conventions in detail.

Worked Problems

Frequently Asked Questions

The SI units physics runs on are seven base units: the second (s), metre (m), kilogram (kg), ampere (A), kelvin (K), mole (mol) and candela (cd). Since 2019 each one is fixed by an exact constant of nature, and every other unit — newton, joule, volt, pascal — is built from these seven.

In September 1999 a NASA Mars orbiter costing about $125 million vanished into the Martian atmosphere. The hardware worked. The navigation maths worked. One team had supplied thruster impulse data in pound-force seconds while the flight software expected newton seconds.

That is the case for a single agreed system of measurement, made in one very expensive sentence. Physics is never just numbers — it is numbers welded to units, and the SI is the welding standard the scientific world agreed to share.

What Are SI Units in Physics?

SI units are the international system of measurement used in physics, built from seven base units and defined by seven exact constants of nature. SI stands for Système International d’Unités. Every measurement in a physics paper, exam paper or engineering drawing can be expressed in it.

Think of it as a language with seven letters. You cannot spell “pressure” without borrowing from mass, length and time — and once you know how those letters combine, you can read any unit you have never met before.

Two features make the SI different from a random collection of units. It is coherent: multiply and divide base units together and you get the right derived unit with no fudge factor of 60 or 5,280 hiding in the middle. And it is defined rather than stored: the units come from constants of nature, not from objects in a vault.

The 7 SI Base Units and What Each One Measures

The seven SI base units are the second, metre, kilogram, ampere, kelvin, mole and candela, measuring time, length, mass, electric current, thermodynamic temperature, amount of substance and luminous intensity respectively.

Notice what is missing. There is no base unit for force, energy, pressure, voltage or speed — every one of those is assembled from the seven above. The list is deliberately minimal.

The candela is the odd one out, and students often ask why brightness gets its own base unit. It is the only one weighted by human physiology: the same radiant power looks brighter to your eye at green wavelengths than at deep red, and the candela bakes that response in.

How the 2019 Redefinition Rebuilt the SI on Constants of Nature

On 20 May 2019 the SI switched from artefacts and recipes to fixed constants, so every unit is now defined by a number that can never drift. Four base units changed definition that day: the kilogram, ampere, kelvin and mole.

Until then, the kilogram was a lump. A platinum-iridium cylinder held near Paris was the kilogram by definition, and its official copies around the world slowly disagreed with it by tens of micrograms over a century. Nobody could say which one had changed.

The fix inverts the logic. Instead of measuring the Planck constant in terms of the kilogram, the SI fixes h at exactly 6.626 070 15 × 10^-34 J s and lets the kilogram fall out of it. A Kibble balance then realises that definition by comparing mechanical power against electrical power.

Each SI base unit is now defined by fixing the numerical value of one constant of nature.

One practical consequence is easy to miss: the speed of light is no longer something we measure. It is a defined number, so measuring “the speed of light” now really means calibrating your metre.

What is still changing

The second is next in line. Optical clocks now beat caesium clocks by orders of magnitude in stability, and the international timekeeping community has been working towards a new definition based on an optical transition, with roughly 2030 as the working target. Nothing you calculate today changes — the numerical value of the second is designed to carry over.

SI Derived Units: How Every Other Unit Is Built

A derived unit is any SI unit made by multiplying or dividing base units together, and 22 of them are important enough to have their own name and symbol. Force, energy and power are the three that matter most in an introductory course.

• N — newton, the SI unit of force (derived)
• kg — kilogram, mass (base unit)
• m — metre, length (base unit)
• s — second, time (base unit)

Read that as a sentence, not a formula: one newton is the force that gives one kilogram an acceleration of one metre per second squared. The unit is Newton’s second law, written in shorthand.

• J — joule, the SI unit of energy, work and heat (derived)
• N · m — one newton of force acting through one metre
• kg · m² / s² — the same thing reduced to base units only

• W — watt, the SI unit of power (derived)
• J — joule, energy transferred
• s — second, the time taken

Here is the table worth bookmarking. The last column is the one examiners love, because reducing a unit to base units is how you check whether an equation can possibly be right.

Some derived units have no special name at all, and that is fine. Speed is metres per second, density is kilograms per cubic metre, and neither needs a fancier label to be perfectly SI.

SI Prefixes: From Quetta to Quecto

SI prefixes are multipliers that scale any unit by a power of ten, and there are 24 of them, running from quetta (10³⁰) down to quecto (10^-30). Four were added in November 2022 — the first expansion since 1991, driven mostly by data storage.

SI prefixes let one base unit cover everything from a proton to the Solar System.

The kilogram prefix trap

The kilogram is the only base unit that already contains a prefix, and that creates a rule people trip over constantly. Prefixes attach to the gram, never to the kilogram.

So a millionth of a kilogram is 1 milligram (mg), not 1 microkilogram. There is no such thing as a “µkg” — write mg and move on.

How to Convert Between Units Without Losing Marks

To convert a unit, multiply by a fraction equal to 1 whose top and bottom are the same physical quantity written in different units, then cancel. It is the only method you need, and it never lets you multiply when you should divide.

• value in old unit — the number you were given, with its unit attached
• conversion factor — a ratio equal to 1, such as (1000 m / 1 km) or (1 h / 3600 s)
• value in new unit — the same physical quantity, relabelled

Try it on 90 km/h. Write it as 90 km/h × (1000 m / 1 km) × (1 h / 3600 s); the km cancel, the h cancel, and 90 000 / 3600 leaves 25 m/s. If you had accidentally inverted a factor, the leftover units would be nonsense and you would catch it immediately.

In practice, most exam mistakes are not conversion errors at all — they are conversions that were never done. Areas and volumes are the classic ambush: 1 cm² is 10^-4 m², not 10^-2 m², because the factor gets squared along with the unit.

SI units versus everything else

Plenty of perfectly legal units are not SI. Some are accepted for use alongside it — the litre, the hour, the tonne, the degree of angle — and others belong to older systems that refuse to die.

Torque is where this bites hardest in real life, because workshop manuals switch systems without warning; our Nm to ft-lb Converter handles that swap in both directions if you would rather not trust a hurried mental factor of 1.356.

How to write SI units correctly

Marks get lost on notation, not just numbers. These are the rules that examiners and journal editors actually enforce:

• Leave a space between number and symbol: 5 kg, not 5kg.
• Never pluralise a symbol: 5 kg, not 5 kgs.
• No full stop after a symbol unless the sentence ends there.
• Symbols named after a person take a capital, but the written-out name does not: N and newton, Pa and pascal, W and watt.
• The unit of mass is kg, never Kg — a capital K means kelvin.
• Do not mix names and symbols: write metres per second or m/s, never metre/s.
• Prefix and unit are joined with no space and no hyphen: km, mA, GPa.
• Only one prefix per unit: nm, not mµm.
• Degrees Celsius take a space before the symbol (25 °C), but degrees of angle do not (25°).

Real-World Examples of SI Units in Action

SI units are not an exam formality — they are the reason satellites, medicines and semiconductors work across borders. Four examples show how far the system reaches.

GPS positioning

Your phone finds itself by timing radio signals from satellites. Since the metre is defined through the speed of light and the second through caesium, a timing error of one nanosecond becomes a position error of about 30 cm. Atomic clocks in orbit are not a luxury; they are the unit definition doing its job.

Semiconductor manufacturing

Chip features are quoted in nanometres, and lithography tolerances run to fractions of a nanometre across a 300 mm wafer. Two fabs on opposite sides of the planet can only build interchangeable parts because both trace their length measurements to the same defined metre.

Drug dosing

Clinical doses are given in milligrams per kilogram of body mass, so the mole and the kilogram are doing safety-critical work. Confusing mg with µg is a thousand-fold error, and it is one of the most common medication mistakes in hospitals.

Energy on your electricity bill

Your supplier bills in kilowatt hours while physics works in joules — and 1 kWh is exactly 3.6 MJ. A 2 kW heater running for three hours uses 6 kWh, or 21.6 MJ, which is the same energy expressed in the coherent SI unit.

Common Misconceptions About SI Units

“SI is just another name for the metric system”

Not quite. The metric system is a family of decimal systems going back to the 1790s, including the older CGS system built on centimetres and grams. The SI is one specific, modern, coherent member of that family, and CGS units such as the erg, dyne and gauss are not SI.

“The kilogram is a metal cylinder in Paris”

It was, until 20 May 2019. The international prototype was formally retired that day and the kilogram is now defined through the Planck constant. The cylinder still exists as a historical object, but it no longer defines anything.

“Kilograms measure weight”

Kilograms measure mass; weight is a force measured in newtons. A 70 kg astronaut still has 70 kg of mass in orbit, but a bathroom scale would read zero because it senses force, not mass. If that distinction is shaky, our guide to weight versus mass works through it properly.

“You must never use degrees Celsius in physics”

The degree Celsius is a legitimate SI derived unit with a special name. The catch is that only kelvin works for absolute temperatures inside formulas such as pV = nRT. Temperature differences are safe in either, since a change of 1 °C is exactly a change of 1 K — a point our comparison of heat versus temperature unpacks further.

How SI Units Relate to Formulas and Dimensional Analysis

Units let you audit an equation before you trust it, because both sides of a correct physical equation must reduce to identical base units. This is dimensional analysis, and it is the fastest error check in physics.

Take v² = u² + 2as. The left side is (m/s)² = m²/s². On the right, 2as is m/s² × m = m²/s². Both sides match, so the equation survives the check — and if a stray factor of time had crept in, it would not have.

The same trick works in reverse. If you know that force is measured in kg · m/s², you can reconstruct half of Newton’s second law from the unit alone, which is handy when a formula sheet goes missing.

Where units really earn their keep is the sanity check on magnitude. A student who calculates a car’s kinetic energy as 3 × 10^-4 J has almost certainly left mass in grams; the unit is right, the number is absurd, and only the second observation saves the answer.

Every equation on our physics formulas reference obeys this rule, and so does every derived unit above — the ampere quietly underwrites electric current, the coulomb, the volt and the ohm in turn.

Worked Problems

Frequently Asked Questions

Two authoritative references are worth bookmarking if you need the primary source: the BIPM page on the SI base units, which is the body that maintains the system, and the NIST guide to SI units for definitions, prefixes and writing conventions.

Physics constants are fixed quantities that appear in physical laws and take the same value everywhere in the universe. Seven of them now define the SI exactly, including the speed of light c = 299,792,458 m/s and the Planck constant h = 6.62607015 × 10^-34 J s. Others, such as the gravitational constant G, must still be measured.

On 20 May 2019, a polished cylinder of platinum-iridium sitting in a vault outside Paris quietly stopped being the kilogram. It had defined mass for 130 years. Its replacement is a number: the Planck constant, fixed forever at 6.62607015 × 10^-34 J s.

That swap tells you what these numbers really are. A constant is not trivia to memorise before an exam — it is the anchor that ties an equation to the physical world, and increasingly it is the definition of the unit itself.

What Are Physics Constants?

Physics constants are quantities whose value does not change with time, place, or the experiment being run, and which appear in the equations describing how nature behaves. Some are fixed by definition; others are measured, and carry an uncertainty.

That distinction matters more than most textbooks admit. Lump them together and you end up quoting g to nine decimal places, or treating G as though it were known as precisely as c. Neither is true.

It helps to sort them into four honest categories:

• Defining constants. Exact by international agreement, with zero uncertainty. c, h, e, k and N_A are in this group.
• Derived exact constants. Built from defining constants by pure arithmetic, so also exact — though irrational. R and the Stefan-Boltzmann constant σ sit here.
• Measured constants. Known only as well as the best experiment allows. G is the classic case.
• Conventional or conditional values. Agreed for convenience, or true only under stated conditions. Standard gravity g and the speed of sound in air both belong here.

Only the first two are genuinely universal in the strict sense. The last group is the one students trip over, and we will come back to it.

Physics Constants Reference Table

The table below lists the constants you will meet in school and first-year university physics, with the CODATA 2022 values published by NIST, SI units, and whether each one is exact or measured.

Trailing dots mean the decimal expansion never terminates, but the value is still exact. Digits in brackets are the standard uncertainty in the final two figures, so 6.67430(15) means 6.67430 ± 0.00015.

If you need constants beyond this working set, NIST publishes the full CODATA list as a one-page wall chart.

Why the SI Is Now Built on Seven Defining Constants

Since 20 May 2019, every SI base unit has been defined by fixing the numerical value of a constant of nature rather than by a physical artefact, as set out in the BIPM definition of the SI. The kilogram now follows from the Planck constant; the metre follows from the speed of light.

Think of it as changing the reference from a ruler in a drawer to a property of the universe. Anyone with the right apparatus can rebuild the kilogram in Nairobi or Nagoya and get the same answer, because the definition travels as a number rather than a lump of metal.

The seven defining constants of the SI and the base unit each one fixes.

Notice what this does to precision. Once a value is fixed by decree, its uncertainty is zero forever, and the uncertainty moves instead into how well we can realise the unit in a laboratory.

The Seven Constants You Actually Use

Five constants carry most of the load in school and first-year physics — g, c, h, G and R — with the Coulomb constant and the speed of sound close behind. Here is what each one does, and the trap that comes with it.

g — Standard Gravity

Standard gravity is the conventional acceleration of free fall near Earth’s surface, fixed at exactly 9.80665 m/s² by the 3rd General Conference on Weights and Measures in 1901.

• W — weight, the gravitational force on the object, in newtons (N)
• m — mass, in kilograms (kg)
• g — gravitational field strength, in newtons per kilogram (N/kg), numerically equal to the free-fall acceleration in m/s²

Here is the catch: g is not a constant of nature at all. Real local gravity runs from roughly 9.78 m/s² at the equator to about 9.83 m/s² at the poles, because Earth spins and bulges.

In practice, use 9.81 m/s² unless a question specifies otherwise. The 9.80665 figure is a legal convention for trade and calibration, not a measurement of your particular hillside.

c — The Speed of Light in Vacuum

The speed of light in vacuum is exactly 299,792,458 m/s, and has been since 1983, when the metre was redefined in terms of it.

• E — energy, in joules (J)
• m — mass, in kilograms (kg)
• c — speed of light in vacuum, in metres per second (m/s)

That exactness is not a boast about measurement. It is a definition: we stopped measuring c and started using it to define length, so the metre is now whatever distance light covers in 1/299,792,458 of a second.

h — The Planck Constant

The Planck constant relates a photon’s energy to its frequency, and is exactly 6.62607015 × 10^-34 J s.

• E — photon energy, in joules (J)
• h — Planck constant, in joule seconds (J s)
• f — frequency, in hertz (Hz)

Because h is tiny, quantum effects stay hidden at everyday scales — a single green photon carries only about 4 × 10^-19 J. If you are converting between wavelength, frequency and energy repeatedly, the photon energy calculator handles the unit juggling for you.

G — The Gravitational Constant

The gravitational constant sets the strength of gravity between any two masses, with a CODATA 2022 value of 6.67430(15) × 10^-11 m³ kg^-1 s^-2.

• F — gravitational force, in newtons (N)
• G — gravitational constant, in m³ kg^-1 s^-2 (equivalently N m² kg^-2)
• m₁, m₂ — the two masses, in kilograms (kg)
• r — separation between their centres, in metres (m)

G is the embarrassment of precision physics. We know the electron’s magnetic moment to about one part in a trillion, yet G is pinned down only to 22 parts per million, because gravity is far too weak to shield from everything else.

It is also the constant students most often confuse with g. They are not related by a shortcut; you can compute the gravitational force between any two objects with the gravitational force calculator and see how different the scales are.

R — The Molar Gas Constant

The molar gas constant links pressure, volume, amount of substance and temperature for an ideal gas, and equals exactly 8.314462618… J mol^-1 K^-1.

• p — pressure, in pascals (Pa)
• V — volume, in cubic metres (m³)
• n — amount of substance, in moles (mol)
• R — molar gas constant, in J mol^-1 K^-1
• T — absolute temperature, in kelvin (K)

R is not fundamental in its own right. It is simply the Boltzmann constant scaled up to one mole, R = N_Ak, which is why it inherited exactness the moment both of those were fixed.

Keep T in kelvin and p in pascals and the units take care of themselves; the ideal gas law calculator is useful for checking a rearrangement you are unsure about.

k_e — The Coulomb Constant

The Coulomb constant sets the strength of the electrostatic force and equals 8.98755179 × 10⁹ N m² C^-2, usually rounded to 8.99 × 10⁹.

• F — electrostatic force, in newtons (N)
• k_e — Coulomb constant, equal to 1/(4πε₀), in N m² C^-2
• q₁, q₂ — the two charges, in coulombs (C)
• r — separation, in metres (m)

Compare k_e with G and the gulf between the two forces becomes obvious: one is around 10⁹, the other around 10^-11. Electrostatics beats gravity by roughly twenty orders of magnitude for everyday particles.

v — The Speed of Sound in Air

The speed of sound in dry air is about 343 m/s at 20 °C, and it changes with temperature rather than with pressure.

• v — speed of sound in dry air, in metres per second (m/s)
• 331.3 — the speed at 0 °C, in m/s
• T — air temperature, in degrees Celsius (°C)

This is the most conditional entry on the page. Quote 343 m/s without stating the temperature and you have quoted a number, not a constant.

Which Physics Constants Are Exact, and Which Are Measured?

Five constants are exact because the SI defines them: c, h, e, k and N_A. Everything else is either arithmetic built from those, or a genuine measurement carrying an uncertainty.

Read the last two rows together and the oddity jumps out. We know the mass of a proton roughly a hundred thousand times more precisely than we know the strength of the force holding the solar system together.

The values themselves are also wildly spread out, which is worth seeing rather than being told.

Physics constants plotted on a logarithmic scale, from the Planck constant to the Avogadro constant.

A useful sanity check follows from that picture: if a calculation involving h returns something near 1, you have almost certainly dropped a power of ten somewhere.

Real-World Examples of Physics Constants at Work

Physics constants are not confined to exam papers; they are wired into technology you use daily. Five cases make the point.

• Satellite navigation (c). Your position is worked out from signal travel times. Light covers about 30 cm in a nanosecond, so a clock error of a few nanoseconds becomes a metre of error on the ground.
• LED lighting and screens (h). The colour of an LED is set by the photon energy it emits. A blue LED at 450 nm puts out photons of about 2.76 eV, and h is the conversion factor that gets you there.
• Kitchen and laboratory scales (g). A scale measures force, then divides by g to display a mass. Ship the same scale from the equator to the Arctic without recalibrating and a 70 kg reading drifts by roughly 0.36 kg.
• Tyre pressure and weather balloons (R). Both are the ideal gas law in disguise: warm the gas and either the pressure or the volume has to give.
• Judging a thunderstorm (v). Count the seconds between flash and bang. Three seconds at 343 m/s puts the strike about a kilometre away.

Common Misconceptions About Physics Constants

Four errors account for most of the marks lost on this topic, and the first is by far the most expensive.

Mistake 1: Treating g and G as the Same Thing

They are different quantities with different units and different natures. G is a universal constant in m³ kg^-1 s^-2; g is a local field strength in N/kg that changes depending on where you stand.

The two connect only through a specific body: g = GM/R², where M and R are that planet’s mass and radius. Confusing them also feeds the older confusion between weight and mass, since weight is the thing that changes when g does.

Mistake 2: Assuming g Is 9.81 Everywhere

Local gravity varies by about half a percent across Earth’s surface, from roughly 9.78 m/s² at the equator to 9.83 m/s² near the poles. Altitude and local rock density shift it further.

Use 9.81 m/s² as a working value, but do not report an answer to five significant figures on the back of it.

Mistake 3: Thinking c Is Still Being Measured More Precisely

It is not, and it cannot be. Since 1983 the metre has been defined from the speed of light, so measuring c more accurately now just measures your ruler.

Any experiment that appears to time light more precisely is really calibrating a length standard.

Mistake 4: Treating R and k as Unrelated

They are the same physics at two different scales. The Boltzmann constant k applies per particle, the gas constant R applies per mole, and R = N_Ak connects them exactly.

Use R when you are counting in moles and k when you are counting individual molecules — mixing them up is a factor of 6 × 10²³ error, which is hard to miss.

How Physics Constants Relate to Formulas, Units and Uncertainty

A constant only means something once it is attached to an equation and a set of units. That is why this page pairs naturally with the wider toolkit.

• Formulas. Every constant here is the fixed term in some relationship; the grouped guide to physics formulas shows where each one slots in.
• Thermodynamics. R does its main work in the ideal gas law, where keeping temperature in kelvin is the whole battle.
• Quantum physics. h is the bridge between frequency and energy, worked through step by step in the guide to photon energy.
• Electrostatics. k_e exists only inside Coulomb’s law, and its enormous size is why static shocks are so easy to generate.

Uncertainty is the thread running through all of it. Your answer can never be more precise than the least precise number you fed in, and with gravity that number is almost always G.

Worked Problems

Frequently Asked Questions

Key Takeaways

• Physics constants split into exact defining values, exact derived values, measured values, and conventional or conditional values.
• c, h, e, k and the Avogadro constant are exact by definition, and together with two others they now define every SI base unit.
• G is the least precisely known constant here, at about 22 parts per million.
• Lowercase g is not a universal constant; it varies from roughly 9.78 to 9.83 m/s^2 across Earth’s surface.
• Always carry units through the algebra, and never report more significant figures than your least precise constant allows.

The de Broglie wavelength is the wavelength associated with any moving particle, equal to Planck’s constant divided by the particle’s momentum (λ = h / mv). Every object with mass and velocity has one. It is large enough to measure for electrons and neutrons, but unimaginably small for everyday objects like a cricket ball.

Fire a stream of electrons at a thin crystal and they do something no particle should do. Instead of arriving as a scatter of dots, they pile up in bright rings and dark gaps — the unmistakable signature of a wave passing through a grating.

That pattern is not a quirk of electrons. It is a property of everything that moves, and a young French PhD student worked out the rule for it in 1924 with one short equation.

What Is the De Broglie Wavelength?

The de Broglie wavelength is the wavelength a moving particle behaves as if it has, found by dividing Planck’s constant by the particle’s momentum. Louis de Broglie proposed it in his 1924 doctoral thesis, and it applies to electrons, neutrons, atoms, molecules and, in principle, you.

Physics already accepted that light — obviously a wave — sometimes behaves like a stream of particles. De Broglie asked the obvious question in reverse. If waves can act like particles, why shouldn’t particles act like waves?

The idea sounds like a philosophical flourish. It isn’t. It makes a hard numerical prediction: give a particle a momentum, and its wavelength is fixed. Nothing is left to interpretation.

The De Broglie Wavelength Formula

The de Broglie wavelength formula is Planck’s constant divided by momentum, written as lambda equals h over mv.

Because momentum p = m·v, the same equation is often written more compactly:

Since 2019 the Planck constant has been fixed by definition rather than measured — its value is exactly 6.62607015 × 10^-34 J·s, as published in NIST’s CODATA tables. The kilogram is now defined from it.

Two rearrangements worth memorising

Exam questions rarely hand you the speed directly. Far more often you get a kinetic energy or an accelerating voltage, so these two forms save real time:

Here K is kinetic energy in joules, q is the particle’s charge in coulombs, and V is the accelerating potential difference in volts. Both follow from p = sqrt(2mK), which is just kinetic energy rearranged for momentum.

If you only need the number and not the algebra, our De Broglie Wavelength Calculator takes a mass and a speed and returns the wavelength with the working shown, which makes it a fast way to sanity-check your own answers.

How the De Broglie Wavelength Works

De Broglie’s derivation starts with light and then makes one bold leap. Follow it in three steps.

Step 1 — start with a photon. Einstein and Planck had already established that a light wave of frequency f carries energy in packets of E = hf, the basis of photon energy.

Step 2 — give the photon momentum. Relativity gives a massless particle p = E/c. Substituting E = hf and c = fλ makes the frequency cancel, leaving p = h/λ, or λ = h/p.

Step 3 — drop the assumption of masslessness. De Broglie’s leap was to claim λ = h/p is not a fact about light at all. It is a fact about momentum, so it must apply to an electron exactly as it applies to a photon.

Nothing in step 3 was justified when he wrote it. It was, in the most literal sense, a hypothesis — and three years later the electrons settled it.

The experiment that proved it

In 1927, Clinton Davisson and Lester Germer at Bell Labs fired low-energy electrons at a nickel crystal and found scattering peaks exactly where wave diffraction predicted them. Their 54 eV electrons behaved like a wave of roughly 0.165 nm.

Plug 54 V into the formula and you get 0.167 nm. That agreement, from an experiment that began as an accident after a piece of glassware imploded, is documented in the American Physical Society’s account of the Davisson–Germer landmark. George Paget Thomson found the same effect in thin films weeks later.

De Broglie Wavelength of an Electron

For an electron accelerated from rest through a potential difference V, the de Broglie wavelength in nanometres is 1.226 divided by the square root of V in volts. It is the single most useful shortcut in this topic.

The constant 1.226 is not arbitrary. It is h / sqrt(2·m·e) converted to nanometres, using the electron mass and the elementary charge — so the shortcut is the full formula with the constants pre-multiplied.

Some quick values: 1 V gives 1.23 nm, 100 V gives 0.123 nm, and 10,000 V gives 0.0123 nm. Every hundredfold rise in voltage shrinks the wavelength by a factor of ten.

In practice, this is why electron microscopes exist. Visible light stalls at roughly 200 nm of resolving power, but a 100 kV electron carries a wavelength near 3.7 pm — about 50,000 times shorter.

Real-World Examples of De Broglie Wavelength

The formula is trivial. What surprises people is the spread of results — twenty-five orders of magnitude between an electron and a walking adult.

De Broglie wavelengths on a log scale. Only particles whose wavelength reaches atomic spacing can be made to diffract.

Where the wave actually shows up

Neutron diffraction is the clearest payoff. Because a room-temperature neutron carries a wavelength close to atomic spacing, firing neutrons at a crystal maps where its atoms sit — the routine method for locating hydrogen atoms that X-rays largely miss.

The most striking demonstration came in 1999, when a Vienna group sent C60 fullerene molecules — 60 carbon atoms, 720 atomic mass units — through a nanofabricated grating. Travelling at 220 m/s, each molecule had a wavelength of 2.5 pm and produced clean interference fringes.

That wavelength is hundreds of times smaller than the molecule itself. Experiments since 2019 have pushed this to molecules beyond 25,000 atomic mass units, built from up to 2,000 atoms.

Why You Never See the Wavelength of a Cricket Ball

Everyday objects do have a de Broglie wavelength, but it is far too small to produce any observable effect. Planck’s constant sits in the numerator at 10^-34, and everyday momentum sits in the denominator at around 1 — so the answer lands near 10^-34 m.

Set that against something small. A proton is about 10^-15 m across, which makes the cricket ball’s wavelength roughly nineteen orders of magnitude smaller than a proton.

To diffract a wave you need an aperture comparable to its wavelength. There is no aperture that small, and there never will be — the ball’s wavelength is only about six times the Planck length.

So the classical world isn’t exempt from quantum mechanics. It just carries too much momentum for anyone to notice.

Common Misconceptions About the De Broglie Wavelength

“Only electrons have a de Broglie wavelength”

Every object with momentum has one. Electrons simply have small enough momentum for the wavelength to be measurable. A bowling ball obeys the same equation and gets an unmeasurable answer.

“The particle wiggles along a wavy path”

Nothing physically oscillates in space. The wave is a probability amplitude — it tells you where the particle is likely to be found, which is why a single electron sent through a double slit still lands as one dot.

“Faster particles have longer wavelengths”

The relationship is inverse. Speed sits in the denominator, so doubling a particle’s speed halves its de Broglie wavelength. This one costs marks in exams more than any other slip in the topic.

“The formula applies to photons too”

Use λ = h/p for a photon and you are fine. Use λ = h/mv and you are dividing by zero, because a photon has no mass. Photon wavelength comes from E = hf instead.

How the De Broglie Wavelength Relates to Other Ideas

It explains Bohr’s orbits

Bohr had assumed that angular momentum in a hydrogen atom comes in multiples of h/2π, with no reason why. De Broglie supplied one: an electron orbit is stable only when a whole number of its wavelengths fits around the circumference.

Substitute λ = h/mv and Bohr’s rule mvr = nh/2π drops straight out. An orbit with a non-integer fit would interfere with itself and cancel, which is exactly why the electron cannot spiral into the nucleus.

A stable Bohr orbit is one where the electron’s matter wave closes on itself: 2πr = nλ.

It completes wave–particle duality

The photoelectric effect showed light behaving as particles. The de Broglie wavelength showed matter behaving as waves. Together they close the loop: the wave and particle descriptions are two views of one thing.

It needs a correction at high speed

Once a particle approaches a noticeable fraction of the speed of light, momentum is no longer simply mv and special relativity takes over. Use p = γmv, where γ is the Lorentz factor.

The error is not academic. A 100 kV electron microscope beam moves fast enough that the non-relativistic answer is wrong by about 5%.

Momentum is doing all the work

Notice that mass and speed never appear separately — only as their product. Two particles with identical momentum share a wavelength no matter how the mass and speed are divided between them.

Worked Problems

Every step is shown, with units carried through. Work them yourself first, then check.

Frequently Asked Questions

Capacitance is the amount of electric charge a component stores for every volt of potential difference across it, defined by C = Q/V and measured in farads. One farad equals one coulomb per volt. A charged capacitor also stores energy equal to half the capacitance multiplied by the voltage squared.

Press the shutter on a camera with a built-in flash and you sometimes hear a faint rising whine before it fires. That is a capacitor filling up. The battery cannot dump its energy fast enough to make a bright flash, so it trickles charge into a capacitor for a second or two, and the capacitor releases the lot in about a thousandth of a second.

The numbers are worth sitting with. A typical AA cell holds roughly 10 kJ of energy; a camera flash capacitor holds around 9 J — about a thousand times less. Yet the capacitor wins, because it can hand that energy over thousands of times faster. Capacitance is the property that decides how much charge a component can park, and how much energy comes back out.

What Is Capacitance?

Capacitance is the charge a conductor stores for every volt applied to it, written C = Q/V. A component with a large capacitance accepts a lot of charge without its voltage climbing much; a small one fills up almost immediately.

Think of a spring. Push on a stiff spring and it barely moves; push on a floppy one and it travels a long way for the same force. Voltage is the push, stored charge is the movement, and capacitance is the floppiness — how much charge you get per volt of push.

The precise definition depends slightly on what you are measuring:

• An isolated conductor — C is the charge on it divided by its potential relative to a point infinitely far away.
• A capacitor (two conductors separated by an insulator) — Q is the magnitude of the charge on each plate, and V is the potential difference between them.

The second case is the one you meet in circuits, and it hides a subtlety worth flagging early. A “charged” capacitor holds +Q on one plate and -Q on the other, so its total charge is zero. It has not stored charge so much as pulled charge apart and held it there.

The Capacitance Formula: C = Q/V

The defining formula for capacitance divides the charge stored on one plate by the potential difference across the capacitor.

Rearranged, the same relationship gives you Q = CV and V = Q/C. If you want the arithmetic done for you while you check your own working, our Capacitance Calculator solves for capacitance, charge or voltage and returns the stored energy alongside.

Here is the point that trips people up. Capacitance is not set by Q or V — it is fixed by the capacitor’s geometry and the insulator inside it. Double the voltage and you double the charge; the ratio between them does not budge.

The Parallel-Plate Formula

For the simplest capacitor — two flat plates facing each other — capacitance follows directly from the geometry:

• ε₀ — the vacuum permittivity, 8.854 × 10^-12 farads per metre (F/m)
• ε_r — relative permittivity (dielectric constant) of the insulator, a dimensionless number
• A — the overlapping area of one plate, in square metres (m²)
• d — the separation between the plates, in metres (m)

This formula assumes the plates are large compared with their spacing, so the field between them is uniform and edge effects are negligible. Real capacitors deviate slightly, but for teaching and for most design work it is accurate enough.

How Much Energy Does a Capacitor Store?

A charged capacitor stores energy equal to half the charge times the voltage — not the full QV. The same quantity can be written three equivalent ways, and which one you reach for depends on what you already know.

• U — energy stored, in joules (J)
• C — capacitance, in farads (F)
• V — potential difference across the capacitor, in volts (V)
• Q — charge on one plate, in coulombs (C)

Where the Half Comes From

Why half? Because the voltage is not constant while the capacitor charges — it climbs from zero to V as charge piles up.

The very first packet of charge crosses an empty capacitor and does almost no work against the field. The last packet has to fight its way across the full voltage. Averaged over the whole process, each unit of charge crosses an average potential of V/2, which is exactly where the factor of one half comes from.

Formally, moving a small charge dq across the instantaneous voltage q/C costs dW = (q/C) dq. Integrating from 0 to Q gives U = Q²/(2C), and substituting Q = CV recovers the other two forms.

A quick sanity check you can use in an exam: if you ever write U = QV, you have claimed the capacitor charged at full voltage from the very start. It did not.

What Changes a Capacitor’s Capacitance?

Three things set the capacitance of a parallel-plate capacitor: plate area, plate separation, and the insulating material between the plates. Each acts in a way you can reason out physically.

• Larger plate area raises capacitance. More room for charge to spread out means less crowding, so less voltage builds up per unit of charge added.
• Smaller separation raises capacitance. The opposite charges pull on each other more strongly across a narrow gap, holding more charge in place at the same voltage.
• A dielectric raises capacitance. The insulator’s molecules polarise and partly cancel the field between the plates, so the voltage drops and C rises by the factor ε_r.

That third effect is why real capacitors are not just two bits of metal in air. Choosing the right dielectric buys you a factor of thousands.

In practice, electrolytic capacitors reach hundreds of microfarads by a different trick: an oxide layer only a few hundred nanometres thick. Because d sits on the bottom of the formula, making it tiny sends C soaring — which is also why those capacitors have modest voltage ratings and fail loudly when you exceed them.

The lab below lets you move all three variables at once. Change the plate area, the gap and the dielectric, and watch capacitance, stored charge and stored energy respond.

Real-World Examples of Capacitance

Capacitance shows up wherever charge needs to be stored briefly, released quickly, or sensed. Five cases give a feel for the range.

1. Camera Flash

A flash capacitor of roughly 200 µF charged to about 300 V stores around 9 J. The battery takes a second or two to fill it; the xenon tube empties it in about a millisecond, which is what makes the flash bright rather than merely warm.

2. Defibrillator

A defibrillator typically charges a capacitor of about 100 µF to roughly 2,000 V, storing 200 J. Note how strongly the voltage dominates — because U depends on V², doubling the voltage quadruples the energy delivered.

3. Smoothing in a Power Supply

Rectified mains voltage arrives as a lumpy series of humps. A large capacitor across the output charges on each peak and discharges gently between them, flattening the ripple into something a circuit can actually run on.

4. Capacitive Touchscreens

Your phone screen carries a grid of tiny capacitors. A fingertip is conductive enough to alter the capacitance at one node by around a picofarad, and the controller reads that change as a touch — which is why gloves usually fail.

5. Supercapacitors

A 3,000 F supercapacitor rated at 2.7 V stores about 10.9 kJ, roughly 3 watt-hours. That is small next to a battery of the same mass, but it can be charged and discharged in seconds and survives hundreds of thousands of cycles — useful for regenerative braking and backup power.

That last row is worth a second look. The entire planet, treated as an isolated conducting sphere, has a capacitance comparable to a large electrolytic capacitor you could hold between two fingers — a reminder that the farad is an enormous unit.

Common Misconceptions About Capacitance

Myth 1: A Capacitor Stores Charge

A charged capacitor has zero net charge. One plate carries +Q and the other -Q, so what the capacitor really does is separate charge and hold it apart. What it stores is energy, in the electric field between the plates.

Myth 2: A Bigger Capacitance Means More Charge

Only at the same voltage. Capacitance is charge per volt, so a 1 µF capacitor at 100 V holds ten times the charge of a 10 µF capacitor at 1 V. Always check what voltage each component is sitting at before comparing.

Myth 3: The Energy Stored Is QV

It is half that. Because voltage climbs from zero to V during charging, the average voltage each unit of charge crosses is V/2. Writing U = QV instead of ½QV is one of the most common ways to lose marks on this topic.

Myth 4: Capacitance Depends on the Voltage You Apply

It does not. C is fixed by area, separation and dielectric — the geometry of the thing. Raising the voltage raises the stored charge in exact proportion, leaving the ratio Q/V unchanged.

How Capacitance Relates to Charge, Voltage and Current

Capacitance ties together three quantities you have already met: charge, potential difference, and the field that connects them. Each one gives a different handle on the same physics.

The V in C = Q/V is the ordinary potential difference across the plates, and the charge separation that produces it is the same effect at work in everyday static electricity. Between the plates sits a uniform electric field of strength E = V/d — for a 12 V supply across a 0.1 mm gap, that is 120,000 V/m.

Charging is not instantaneous, either. Charge arrives as a current, and since electric current is the rate of charge flow, the capacitor fills at a speed set by whatever resistance is in the way. The product RC is the time constant, the time to reach about 63% of the final voltage.

Capacitors in Series and Parallel

Here is a rule that catches almost everyone: capacitors combine the opposite way round to resistors.

• In parallel — capacitances add: C_total = C₁ + C₂ + … The plate areas effectively combine.
• In series — reciprocals add: 1/C_total = 1/C₁ + 1/C₂ + … The total is always smaller than the smallest one.

If that feels backwards, check it against how series and parallel circuits behave with resistors and the symmetry becomes obvious rather than arbitrary.

Worked Problems

Frequently Asked Questions

Key Takeaways

• Capacitance is charge stored per volt: C = Q/V, measured in farads (1 F = 1 C/V).
• For parallel plates, C = ε₀ε_rA/d — more area and less separation both raise C.
• Stored energy is U = ½CV² = ½QV = Q²/(2C), never the full QV.
• A charged capacitor carries no net charge; it separates charge and stores energy in the field.
• Capacitors add in parallel and combine reciprocally in series — the reverse of resistors.

Time dilation is the difference in elapsed time between two clocks, caused by relative motion or by a difference in gravitational potential. A clock moving at speed v is measured to tick slow by the Lorentz factor: the observed time equals the clock’s own proper time divided by the square root of one minus v squared over c squared.

Somewhere above your head, right now, about thirty atomic clocks are being deliberately run at the wrong speed. They are aboard GPS satellites, and they were built slightly slow on purpose — because once in orbit, they gain roughly 38 microseconds a day on every clock down here.

That is not a manufacturing fudge. It is Einstein, corrected for in hardware, so that your phone can tell you which side of the street you are standing on. Time really does run at different rates for different observers, and we have been measuring it for over sixty years.

What Is Time Dilation?

Time dilation is the effect where a clock is measured to tick more slowly than an identical clock you are holding, because the first clock is either moving relative to you or sitting deeper in a gravitational field.

Here is the part that trips people up. Nothing goes wrong with the moving clock. Its springs are fine, its atoms behave normally, and an astronaut travelling with it sees it tick once per second, forever.

The disagreement is about duration itself — how much time passed between two events. Two observers in relative motion genuinely measure different amounts, and both are right.

Proper time: the one thing you must get straight

Every clock measures its own proper time, written t₀. That is the time between two events as read by a clock that was present at both of them.

Everyone else — anyone who sees that clock move — measures a longer interval. Proper time is always the shortest time anyone measures between two events, and it is always the value on the bottom of the fraction.

The Time Dilation Formula

The time dilation formula relates the time t measured by a stationary observer to the proper time t₀ ticked by the moving clock itself:

That messy denominator appears so often it gets its own name and symbol — the Lorentz factor, γ (gamma):

Every symbol, with its SI unit:

• t — dilated time measured by the observer who sees the clock moving, in seconds (s)
• t₀ — proper time measured by the moving clock itself, in seconds (s)
• v — relative speed between clock and observer, in metres per second (m/s)
• c — speed of light in vacuum, exactly 299,792,458 m/s
• γ — Lorentz factor, a pure number with no units, always ≥ 1

Because γ is never less than 1, t is never smaller than t₀. Moving clocks are measured to run slow — never fast.

You can check any of the numbers below against our Time Dilation Calculator, which takes a proper time and a speed and returns both the dilated time and γ, so you can see how brutally the answer changes in the last decimal places of v.

How big is γ at real speeds?

This table is the fastest cure for the usual intuition that “going fast slows time down a bit”. Below about a tenth of light speed, the effect is essentially nothing.

Sanity check before you trust a calculation: at everyday speeds γ − 1 lands somewhere around 10⁻¹³. If your working shows a noticeable effect for a car or a plane, you have dropped a decimal place — not discovered anything.

The Lorentz factor barely leaves 1 until speeds approach c — which is why time dilation is invisible in everyday life.

How Time Dilation Works: The Light Clock

Time dilation follows from one stubborn fact: every observer measures light travelling at the same speed c, no matter how they are moving. Hold that fixed, and something else has to give — and that something is time.

Imagine the simplest possible clock. Two mirrors face each other, a distance L apart, and a pulse of light bounces between them. One tick is one round trip.

Standing next to it, the light travels straight up and back: a distance 2L, taking t₀ = 2L/c.

Now watch that same clock fly past you at speed v. During one tick, the whole apparatus shifts sideways, so the pulse traces a stretched V — a longer path.

But the pulse cannot travel faster to compensate. Its speed is still exactly c. A longer path at the same speed can only mean one thing: the tick takes longer.

The light clock: with c fixed for every observer, the moving clock’s pulse covers a longer path, so one tick takes longer.

Getting the formula out of the triangle

Apply Pythagoras to half a tick. The vertical side is L, the horizontal side is v·t/2, and the light path is the hypotenuse c·t/2.

Substituting L = c·t₀/2 and rearranging for t gives exactly the time dilation formula. No new assumptions are needed — only constant c and a right-angled triangle.

That is worth sitting with for a moment. One experimental fact about light, plus GCSE geometry, produces the whole result.

Velocity vs Gravitational Time Dilation

There are two distinct causes of time dilation, and mixing them up is the single most common conceptual error in this topic.

Velocity time dilation comes from special relativity and depends on relative speed. Gravitational time dilation comes from general relativity: clocks deeper in a gravitational well run slow compared with clocks higher up.

For weak fields near Earth’s surface, the gravitational shift over a height h is close to gh/c², where g is the gravitational field strength. It is the same g that governs gravitational potential energy — higher potential, faster clock.

Notice the sign clash in that last row. On a GPS satellite the two effects fight each other, and gravity wins by about 38 microseconds a day.

The Twin Paradox, and What Actually Resolves It

The twin paradox is resolved by the fact that only one twin changes inertial frames: the traveller turns around, the stay-at-home twin does not, so their situations are not symmetric and the traveller genuinely returns younger.

Set it up properly first. One twin flies to a star 4.0 light years away at 0.8c and comes straight back; the other stays on Earth.

Earth’s clock records 10 years for the round trip. With γ = 1.667, the traveller’s own clock records only 6 years. She comes home four years younger than her sister — no metaphor, actually younger.

So where is the paradox?

Motion is relative, so the traveller could claim she stood still while Earth flew away and came back. By that logic her sister should be the younger one. Both cannot be right.

The escape is that the two stories are not mirror images. The stay-at-home twin sits in a single inertial frame the whole time.

The traveller does not. To come home she must decelerate and accelerate, switching to a different frame — and she feels it, pressed into her seat.

That turnaround breaks the symmetry. It is not that acceleration magically ages her; it is that her outbound and inbound frames disagree about which distant events are simultaneous, and the switch between them is what the accounting must include.

4 Experiments That Proved Time Dilation Is Real

Time dilation has been confirmed directly by flying atomic clocks around the world, by cosmic-ray muons reaching sea level, by optical clocks moved by a few centimetres, and by the continuous operation of GPS. These are measurements, not thought experiments.

1. Hafele and Keating flew four atomic clocks (1971)

In October 1971 two researchers bought airline tickets for four caesium-beam atomic clocks and flew them around the world — eastward first, then westward — comparing them afterwards against clocks at the US Naval Observatory.

Flying east adds to Earth’s rotation and flying west subtracts, so the two trips shift the velocity term in opposite directions while altitude raises both clocks out of the gravity well. The predictions were genuinely different for the two directions, which makes it a sharp test.

The westward agreement is remarkable: a prediction of +275 ns against a measurement of +273 ns. Both results were published in Science in 1972.

2. Muons that should never reach the ground

Muons are created when cosmic rays strike the upper atmosphere, and they are unstable, with a mean lifetime of about 2.2 microseconds at rest. Travelling at 0.995c, a typical muon covers only about 660 metres in one mean lifetime.

Mount Washington is roughly 1,900 metres above the sea-level detector. Without time dilation, only about 5% of the muons counted at the summit should survive the trip down.

They do not behave that way. Because γ ≈ 10, the muons’ own clocks record only about 0.64 µs for a journey that takes 6.4 µs in our frame, and roughly three quarters of them arrive.

Frisch and Smith measured exactly this at Mount Washington in 1963, extracting a time dilation factor of 8.8 ± 0.8. Later storage-ring work at CERN pushed the same test to γ ≈ 29 at the 0.1% level.

A slip worth avoiding: that 2.2 µs figure is the mean lifetime, not the half-life. The muon half-life is 2.2 × ln2 ≈ 1.52 µs, and swapping them silently wrecks any survival calculation. If you are shaky on the distinction, our guide to half-life in physics sorts it out.

3. Optical clocks raised by 33 centimetres (2010)

You do not need a mountain any more. Physicists at NIST compared two aluminium-ion optical clocks and detected the gravitational shift after raising one of them by just 33 cm.

The same team also measured velocity time dilation at speeds under 10 m/s — jogging pace. NIST’s report on the experiment puts the height effect at roughly 90 billionths of a second over a 79-year lifetime.

Run gh/c² yourself for h = 0.33 m and 79 years and you get 9.0 × 10⁻⁸ s. The back-of-envelope estimate and the world’s best clocks agree.

4. GPS: relativity running continuously since 1978

GPS satellites orbit at about 20,000 km and move at roughly 3.87 km/s. Velocity dilation costs their clocks about 7 µs a day; the weaker gravity up there gains them about 45 µs a day.

The net 38 µs per day is not negligible — multiply it by c and you get an 11 km positioning error accumulating daily. Ohio State’s summary of GPS and relativity notes a fix would be measurably wrong within about two minutes.

Engineers solved it before launch: the onboard oscillators are set to 10.22999999543 MHz so that, once in orbit, they tick at the intended 10.23 MHz.

Common Misconceptions About Time Dilation

“You would feel time slowing down”

You never do. Your own proper time always advances at one second per second, whether you are sitting still or crossing the galaxy at 0.999c.

Time dilation is strictly a comparison between clocks, not a sensation. The traveller notices nothing odd until she gets home and compares calendars.

“It is just clocks malfunctioning”

If moving clocks merely broke, the muons would still decay on schedule and never reach the ground. They arrive because less time genuinely elapsed for them.

Every physical process — decay, chemistry, ageing, thought — slows by the identical factor, because it is the time interval itself that differs.

“It only matters near the speed of light”

GPS satellites crawl along at 0.0013% of light speed, and the effect still breaks navigation within minutes if ignored. NIST measured it at walking pace.

What is true is that the effect is small at low speed, not that it is absent. Precision, not velocity, decides whether you notice.

“Time dilation would let you travel back in time”

It only ever runs one way. A fast traveller can leap far into Earth’s future — fly around for a year at γ = 100 and a century passes back home.

Getting back is another matter. Nothing in the formula reverses the sign, because γ is always ≥ 1.

How Time Dilation Connects to the Rest of Physics

Time dilation is one consequence of a single framework, so it never travels alone. It comes packaged with length contraction, relativistic momentum and mass–energy equivalence.

The full picture — both postulates, the other consequences, and where the theory’s limits lie — is laid out in our guide to special relativity.

The whole edifice rests on c being invariant, which is stranger than it first sounds and is worth reading about on its own in the speed of light.

Push γ high enough and the energy cost explodes too, which is where E = mc² enters: reaching γ = 10 means supplying nine times the particle’s rest energy as kinetic energy. That is why accelerators are expensive, and why starships stay fictional.

Worked Problems

Frequently Asked Questions

Drag force is the resistance a fluid — air, water, or any gas or liquid — exerts on an object moving through it, always acting opposite to the object’s motion. It is calculated with the drag equation: one half the fluid density, times the speed squared, times the drag coefficient, times the frontal area.

Stick your hand out of a car window at 30 km/h and the air barely nudges it. Do the same at 110 km/h and your arm is shoved backwards hard enough to hurt. Same hand, same air — the only thing that changed was speed.

That shove is drag, and it is the reason a lorry burns most of its diesel pushing air rather than moving cargo, the reason a cyclist tucks low, and the reason a bullet is pointed rather than blunt. Get the equation below and you can predict all three.

What Is Drag Force?

Drag force is the backward push a fluid exerts on any object moving through it. It acts along the line of motion, pointing the opposite way, and it exists whether the object is falling, driving, swimming, flying or being pedalled.

Here is the mental picture that makes it click. To move forwards, you have to shove fluid out of the way — and shoving anything takes force.

That fluid pushes back with an equal and opposite force. That is drag.

Two things follow immediately, and both surprise students.

• Drag does not care how heavy you are. Mass appears nowhere in the drag equation. A hollow plastic car and a lead-filled one of identical shape feel identical drag at the same speed.
• Drag does care enormously how fast you are. Double the speed and drag goes up four times, not two.

Physicists call drag a dissipative force: the work it does against you is not stored anywhere recoverable, it is dumped into the fluid as churned-up, warmed-up, swirling air or water. That is energy you paid for and will never get back.

Drag is mostly about what happens behind an object, not in front of it. A wide separated wake is a low-pressure hole the object never stops falling into.

The Drag Force Formula

The drag force formula is F = ½ · ρ · v² · C_d · A, where the drag force in newtons equals one half the fluid density times the square of the speed times the drag coefficient times the frontal area.

Where the ½ρv² Comes From

The group ½ρv² is not an arbitrary fudge — it is the dynamic pressure, the pressure a moving fluid brings with it. The same term sits at the heart of Bernoulli’s principle, which is why the two topics keep bumping into each other.

Read the equation as a sentence and it stops being intimidating: drag = dynamic pressure × area × a shape penalty. Dynamic pressure sets the scale, area sets how much of the flow you intercept, and C_d is the correction factor for how badly your particular shape handles it.

At 30 m/s in air, ½ρv² comes to 551 Pa. Present one square metre of frontal area to that and a perfect flat plate would feel around 700 N — the weight of a heavy adult, made of nothing but air.

Once you have all four inputs in SI units, the arithmetic is quick, and you can check any answer in this article against our Drag Force Calculator, which solves the same equation for force or for speed and shows the substitution step by step.

How Drag Force Works: Pressure Drag and Skin Friction

Drag comes from two physically different mechanisms — pressure (form) drag from the difference in pressure between the front and back of an object, and skin-friction drag from the fluid shearing along its surface. Almost every real object experiences both, in wildly different proportions.

Pressure Drag — the Expensive One

Air piles up in front of a moving object, raising the pressure there. Behind it, the flow cannot follow the shape round a sharp corner, so it separates and leaves a churning low-pressure wake.

High pressure at the front, low pressure at the back — that pressure difference multiplied by the area is a backward force. For a blunt shape like a van, a cube or a cyclist sitting upright, this accounts for the overwhelming majority of the total drag.

Skin-Friction Drag — the Quiet One

Right at the surface, fluid sticks to the object and is dragged along with it. That thin sheared layer, the boundary layer, exerts a tangential force over the whole wetted surface.

On a well-streamlined body — an aerofoil, a submarine hull, a Tour de France skinsuit — the wake is tiny, so skin friction becomes the dominant contribution. This is why competitive swimsuits, aircraft skins and racing yacht hulls obsess over surface texture that would be irrelevant on a lorry.

Why C_d Is Not Really a Constant

Here is the part most textbooks skate over. Whether the boundary layer is smooth (laminar) or chaotic (turbulent) when it separates changes the wake size dramatically — and therefore changes C_d.

The governing quantity is the Reynolds number, Re = vL/ν, comparing inertial to viscous effects. For a smooth sphere, C_d sits near 0.5 for a wide band of everyday speeds, then drops abruptly to roughly 0.1 as Re passes about 3 × 10⁵.

That sudden fall is the drag crisis, and golf-ball dimples exist to trigger it early. Dimples deliberately trip the boundary layer turbulent, which makes it cling further round the back, shrinking the wake. A dimpled ball flies close to twice as far as a smooth one hit identically.

The practical takeaway: a quoted C_d is valid for the conditions it was measured in. Use it near those conditions and it is excellent. Use it four orders of magnitude away in Reynolds number and it is fiction.

Drag Coefficient Values for Common Shapes

Drag coefficients for common shapes range from about 0.04 for a fully streamlined body to about 1.3 for an open parachute — a spread of more than thirty times, all at the same speed and the same frontal area. Shape is the single biggest lever you have.

Every value below uses the frontal (projected) area as the reference area, at ordinary subsonic speeds. Values marked as NASA wind-tunnel figures come from NASA Glenn’s Shape Effects on Drag reference; the rest are standard textbook and industry figures, quoted as ranges because that is honestly how well they are known.

Why Engineers Quote C_dA, Not C_d

C_d on its own is a trap when comparing real vehicles, because a low coefficient on a huge frontal area still means a lot of drag. Engineers therefore multiply the two together into a single number: the drag area, C_dA, measured in m².

• Saloon car, C_d 0.30 × 2.2 m² → C_dA ≈ 0.66 m²
• Large SUV, C_d 0.35 × 2.8 m² → C_dA ≈ 0.98 m²
• Cyclist upright, C_d 1.1 × 0.50 m² → C_dA ≈ 0.55 m²
• Cyclist in a tuck, C_d 0.88 × 0.36 m² → C_dA ≈ 0.32 m²

Read that list again and notice something: an upright cyclist has a drag area close to a car’s. The bike is not what makes you slow — you are.

Why Drag Grows with Speed Squared — and Power with Speed Cubed

Drag grows with the square of speed because the term v² sits in the equation, so doubling your speed multiplies drag by four. The power needed to overcome that drag grows with the cube of speed, so the same doubling multiplies your power requirement by eight.

The second half of that sentence is the one that actually runs the world, and almost nobody is taught it.

Power is force times velocity. Substitute the drag equation and you get:

Two factors of v come from the drag itself; the third comes from having to deliver that force over more metres per second. The result is a wall that gets steeper the harder you push at it — and it explains why power in physics is the honest currency for anything that has to cruise.

The power curve is flat where you do not care and vertical where you do. That is the whole story of top speed.

Put real numbers on it. Take a car with C_d = 0.30 and A = 2.2 m² in ordinary air:

• At 100 km/h (27.8 m/s): drag ≈ 312 N, needing about 8.7 kW just to push air aside.
• At 120 km/h (33.3 m/s): drag ≈ 449 N, needing about 15.0 kW.

A 20 % speed increase raised drag by 44 % and aerodynamic power by 73 %. In practice this is exactly why fuel economy falls off a cliff above roughly 90 km/h, and why the single most effective fuel-saving action available to any driver is easing off by 10 km/h.

Real-World Examples of Drag Force

Drag force shows up wherever something moves through air or water — and in most of those cases it is the dominant force the machine or athlete is fighting. Five examples, each with its own lesson.

1. Cars at Motorway Speed

At 108 km/h, a typical saloon (C_d 0.30, A 2.2 m²) faces about 364 N of drag and burns roughly 11 kW overcoming it. That is why manufacturers fight over the second decimal place of C_d — and why a roof box, which barely changes the car’s weight, can wreck its economy.

2. Cyclists

Air resistance dominates cycling above about 15 km/h, and by racing speeds it is nearly the whole battle. A rider in a tuck (C_dA ≈ 0.32 m²) at 36 km/h faces around 19 N of drag, needing about 194 W of aerodynamic power.

Sit up straight and the drag area jumps to roughly 0.55 m². Same legs, same bike, nearly 70 % more air to shift — which is the entire reason a peloton exists.

3. Articulated Lorries

A tractor-trailer presents around 10 m² of frontal area at C_d ≈ 0.8. At 90 km/h that is roughly 3.1 kN of drag and about 77 kW of engine power spent on air alone.

Fit cab fairings and side skirts to bring C_d down to 0.65 and you save about 14 kW — nearly 19 % — without touching the engine. Over a fleet’s annual mileage, that is the difference between profit and loss.

4. Footballs and Cricket Balls

Here is the number that surprises everyone. A struck football (0.43 kg, 22 cm across) travelling at 30 m/s meets roughly 5.2 N of drag, while its own weight is only about 4.2 N.

The air pushes back on that ball harder than gravity pulls it down. Any trajectory you calculate for it using clean projectile motion and no drag will be badly wrong — the real ball lands much shorter and much steeper.

5. Swimmers, Hulls and Anything in Water

Water is about 816 times denser than air. Swap the fluid and hold everything else fixed, and the drag equation says the force multiplies by that same factor.

That single ratio explains why a swimmer moving at 2 m/s is working harder against the water than a cyclist at 2 m/s is against the air, by a margin no amount of technique closes. It also explains why hull design has obsessed shipbuilders for three thousand years, and it depends directly on fluid density.

Common Misconceptions About Drag Force

Four specific beliefs cause most of the wrong answers in exams and most of the bad intuition outside them. Each one is worth correcting properly.

Misconception 1: “Heavier objects experience more drag”

Mass does not appear in the drag equation at all. Two objects of identical shape and size feel identical drag at identical speed, whether one is polystyrene and the other is lead.

What mass changes is the consequence of that drag — the acceleration it produces, via Newton’s second law. The heavier object shrugs the same force off more easily. That is a different statement, and mixing the two up is the most common slip we see.

Misconception 2: “The drag coefficient is a fixed property of a shape”

C_d is a measured summary of complicated physics, not a material constant. It shifts with Reynolds number, with surface roughness, with the object’s angle to the flow, and near the speed of sound it changes character entirely.

The sphere is the cautionary tale: the same smooth ball can have a C_d of 0.5 or 0.1 depending only on how fast it is going.

Misconception 3: “You can compare any two C_d values directly”

Only if both were measured against the same reference area — and often they are not. Car figures use frontal area, but aircraft figures conventionally use wing planform area, which is far larger.

This is why an airliner’s quoted C_d of around 0.03 does not mean it is ten times slipperier than a good car. NASA’s own drag coefficient reference makes the point explicitly: when you report a C_d, you must state the reference area, or the number means nothing.

Misconception 4: “Drag is just friction with the air”

Only a slice of it is. Surface friction — the skin-friction part — is real, but for the blunt objects most of us care about, the majority of drag is pressure drag from the wake, which has nothing to do with rubbing.

Streamline the tail of a shape without changing its front or its surface at all and drag can fall by a factor of ten. Nothing about the “friction” has changed. The wake has.

How Drag Force Relates to Friction, Terminal Velocity and Fluid Pressure

Drag is one member of a family of resistive forces, and it behaves quite differently from its relatives. Lining them up side by side is the fastest way to stop confusing them.

• Versus dry friction. Surface friction follows F = μN — independent of speed and dependent on the normal force. Drag is the opposite on both counts: fiercely speed-dependent, and completely indifferent to how hard surfaces press together. They sit in different branches of the types of forces taxonomy for good reason.
• Versus terminal velocity. When an object falls, drag grows until it exactly balances weight and the object stops accelerating. That special case has its own full treatment in our guide to terminal velocity, including the rearranged formula and worked skydiver problems.
• Versus fluid pressure. The ½ρv² in the drag equation is dynamic pressure, the same quantity that trades against static pressure in Bernoulli’s principle. Drag and lift are two faces of one pressure field.

One boundary worth knowing: everything above assumes the fast, wake-dominated regime where drag goes as v². For very small or very slow objects — fog droplets, bacteria, a ball bearing sinking through oil — viscosity rules instead and drag becomes proportional to v, described by Stokes’ law rather than the drag equation.

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Angular velocity is the rate at which an object rotates through an angle, equal to the angular displacement divided by the time taken. Its SI unit is the radian per second (rad/s) and its symbol is omega (ω). For steady rotation, angular velocity equals 2π divided by the period, T.

Stand near the middle of a playground roundabout and you can walk it off without much trouble. Sit on the outer rim of that same roundabout, spinning at exactly the same rate, and you will be flung sideways the instant your feet leave the ground.

Nothing about the spin changed — only your distance from the centre. That is why physics needs a quantity describing the whole turning object at once, instead of the speed of one point on it.

What Is Angular Velocity?

Angular velocity is how fast an object turns, measured as the angle it sweeps out per unit of time. Ordinary velocity asks how many metres pass per second; angular velocity asks how many radians are swept per second.

Take the second hand of a clock. It covers a full turn — 2π radians — every 60 seconds, so its angular velocity is 2π ÷ 60 ≈ 0.105 rad/s. The tip of that hand travels far faster than its middle, yet both share the same 0.105 rad/s.

That is the whole idea. A rigid rotating body has one angular velocity, whatever its size and whichever point you choose to look at.

Angular Velocity vs Angular Speed

Angular speed is just the size of the rotation rate — a plain number with units of rad/s. Angular velocity adds a direction, and that direction is a genuine surprise to most students.

It does not point around the circle. It points along the rotation axis, found with the right-hand rule: curl the fingers of your right hand the way the object spins, and your thumb points along ω.

Why the axis? Because it is the only direction in a spinning body that stays put. Every other direction is being swung around, so none of them could label the motion consistently.

Angular velocity relates the swept angle θ to time. The vector ω lies along the rotation axis, set by the right-hand rule.

The Angular Velocity Formula

The angular velocity formula is ω = θ / t: divide the angle turned through by the time taken. It is the rotational twin of the familiar v = d / t.

• ω — angular velocity, in radians per second (rad/s)
• θ — angular displacement, the angle swept, in radians (rad)
• t — time interval, in seconds (s)

When the rotation is steady, there is a shortcut. One complete turn is always 2π radians, so if you know how long a turn takes — the period, T — you already know ω.

• T — period, the time for one full revolution, in seconds (s)
• f — rotation frequency, in hertz (Hz), where f = 1 / T

That factor of 2π is where marks are lost. Angular velocity and rotation frequency describe the same spinning, but their numbers differ by 2π — a point the SI Brochure flags explicitly as a source of error in published work.

The third formula is the bridge to everyday speed, and the one that finally explains the roundabout. Multiply angular velocity by the distance from the axis and you get the tangential speed at that point.

• v — tangential (linear) speed, in metres per second (m/s)
• r — perpendicular distance from the rotation axis, in metres (m)

One condition matters here: ω must be in rad/s. Feed degrees per second or rpm into v = ωr and the answer is simply wrong. If you would rather check a value than grind through the algebra, our Angular Velocity Calculator solves ω = θ / t for any of the three variables and handles the units for you.

Why the SI Unit Is rad/s

A radian is a ratio: arc length divided by radius, metres over metres. It cancels, which makes the radian dimensionless — so rad/s is really just “per second” in disguise.

That is exactly why the unit is still written out in full. Frequency, angular velocity and radioactive activity all reduce to s^-1, and NIST’s edition of the SI Brochure recommends always writing Hz or rad/s rather than s^-1 so the quantities cannot be confused.

How Do You Convert rpm to rad/s?

To convert rpm to rad/s, multiply by 2π and divide by 60 — that is, multiply by roughly 0.10472. To go the other way, multiply rad/s by 60 / 2π, or about 9.5493.

Machines are labelled in revolutions per minute; physics equations demand radians per second. The conversion is a two-part job, and skipping half of it is the single most common slip in rotational problems.

1. Revolutions to radians: multiply by 2π, because one revolution is 2π radians.
2. Minutes to seconds: divide by 60.
3. Combine: ω (rad/s) = rpm × 2π / 60 = rpm × 0.10472.

A worked instance: a drill quoted at 1,200 rpm turns at 1,200 × 0.10472 = 125.7 rad/s. Divide by 60 alone and you would get 20 — wrong by a factor of 2π, roughly six times too small.

Degrees per second needs converting too. Multiply deg/s by π / 180 (about 0.01745) to reach rad/s; one radian per second is about 57.3 deg/s.

How Angular Velocity Works

Angular velocity works because arc length, radius and angle are locked together by s = rθ, so a single rotation rate fixes the speed of every point on the body. Differentiate that relationship and v = ωr falls straight out.

Follow it step by step. A point at distance r sweeping an angle θ travels an arc of length s = rθ, provided θ is in radians.

Divide both sides by the time taken. The left-hand side becomes arc length per second — the tangential speed v. The right-hand side becomes r × (θ / t), which is r × ω.

So v = ωr, and the radius is the only thing separating one point from another. Double the distance from the axis and you double the speed, while ω sits there unchanged.

Every point shares one angular velocity, but tangential speed v = ωr rises in direct proportion to the radius.

This is the mechanism behind almost every rotating machine. Gears, belts and pulleys are all devices for trading angular velocity against radius, and it is the same relationship at the heart of circular motion.

Drag the sliders below to watch it happen: change the radius or the spin rate and see how the tangential speed responds while ω stays put.

Real-World Examples of Angular Velocity

Angular velocity spans an enormous range in ordinary life, from a planet creeping round once a day to a centrifuge rotor screaming at thousands of radians per second. Five cases show the spread.

1. The Earth Spinning on Its Axis

Earth completes one rotation relative to the stars in 23 h 56 min 4 s — not 24 hours. NASA notes that this sidereal day runs 3 minutes 56.55 seconds short of the mean solar day, because Earth also moves along its orbit.

That gives T ≈ 86,164 s, so ω = 2π / 86,164 ≈ 7.29 × 10^-5 rad/s. Tiny — yet at the equator, where r ≈ 6,378 km, it still works out to v = ωr ≈ 465 m/s.

2. A Vinyl Record

An LP turns at 33 1/3 rpm, which is 3.49 rad/s. The needle near the outer edge (r ≈ 0.15 m) moves at 0.52 m/s, but by the final track (r ≈ 0.06 m) that has dropped to 0.21 m/s.

Same ω throughout, less groove passing the stylus every second — which is precisely why the inner tracks of an LP sound worse.

3. A Wind Turbine

A large turbine rotor turns slowly, around 15 rpm, giving ω ≈ 1.57 rad/s. On a 40 m blade, though, the tip is doing v = 1.57 × 40 ≈ 63 m/s — over 220 km/h.

4. A Hard-Disk Drive

A 7,200 rpm platter spins at 754 rad/s, completing a revolution in 8.3 milliseconds. That period sets the drive’s rotational latency: on average the read head waits half a turn, roughly 4 ms, for the right sector to arrive.

5. A Laboratory Centrifuge

At 12,000 rpm a rotor reaches 1,257 rad/s. A sample sitting 8.0 cm from the axis then experiences a centripetal acceleration of ω²r ≈ 1.3 × 10⁵ m/s² — about 13,000 g, which is what drives the separation.

Common Misconceptions About Angular Velocity

Misconception 1: Angular Velocity and Linear Velocity Are the Same Thing

They are not, and conflating them wrecks otherwise correct working. Every point on a rigid body shares one angular velocity; almost none of them share a linear velocity.

On a spinning bicycle wheel, the valve near the hub and the tread at the rim both complete a turn in the same time. The tread simply has further to travel, so it moves faster.

Misconception 2: Converting rpm Just Means Dividing by 60

Dividing by 60 converts minutes to seconds and stops there — it leaves the answer in revolutions per second, not radians per second. You still owe the factor of 2π.

In practice, a missing 2π shows up as an answer that is about six times too small. If a result looks suspiciously modest for something visibly spinning fast, check this first.

Misconception 3: Constant Angular Velocity Means No Acceleration

A body turning at constant ω is accelerating the entire time. Its speed is steady, but its direction changes continuously, and velocity is a vector.

That change of direction is a real acceleration pointing at the axis, of size a = ω²r. It is supplied by a genuine centripetal force — string tension, friction, gravity — and if that force disappears, the object leaves along the tangent.

Misconception 4: Angular Velocity Points Around the Circle

Angular velocity is an axial vector: it lies along the rotation axis, not along the circular path. Only its magnitude, the angular speed, behaves like the scalar most people expect.

This matters as soon as rotations start interacting. Gyroscopes, precessing tops and the stability of a moving bicycle all depend on ω having a fixed direction in space that torques can push against.

How Angular Velocity Relates to Torque, Momentum and Oscillation

Angular velocity sits at the centre of rotational dynamics: change it and you need torque, multiply it by inertia and you get angular momentum, square it and you get rotational energy. Each linear quantity has a rotational partner.

When ω itself changes, the rate of change is the angular acceleration, α = Δω / Δt, measured in rad/s². Producing it takes torque, through τ = Iα — the rotational form of Newton’s second law.

Multiply angular velocity by the moment of inertia and you get angular momentum, L = Iω. Like linear momentum, it is conserved when no external torque acts.

That single fact explains the spinning skater. Pulling her arms in cuts I, so ω must rise to keep L constant — the same physics that makes a collapsing star spin up into a pulsar.

Rotational kinetic energy follows the same pattern: KE = ½Iω², the exact echo of ½mv². The ω in a flywheel is squared, which is why doubling the spin rate quadruples the energy stored.

One caution on vocabulary. In simple harmonic motion the symbol ω means angular frequency, and nothing is physically rotating — it is a bookkeeping device, since ω = 2πf describes how fast the phase advances.

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