Mechanics Formulas Cheat Sheet

Open any mechanics textbook to the back cover and you will find the same thing: a wall of equations, printed small, with no clue about which one to reach for. Most students copy it out, memorise it, and still stall on the first line of a problem.

That stall is almost never a memory failure. It is a matching failure — you know the equations, you just cannot tell which one fits the numbers in front of you. This sheet is arranged to fix that: every formula is grouped by the quantity it produces, every symbol carries its SI unit, and every row that has a calculator on this site links straight to it.

What Are Mechanics Formulas?

Mechanics formulas are the equations describing how forces make objects move, deform, spin and collide. They cover everything from a dropped key to a satellite holding orbit, and they all rest on a handful of conserved quantities and one law relating force to acceleration.

Think of them less as a list to store and more as a set of translation rules. Each formula converts a group of measurable things — a mass, a distance, a time — into something you cannot measure directly, like the force in a cable or the speed at the bottom of a slope.

Mechanics splits neatly into six families, and that split is the whole trick to using the sheet below:

• Kinematics — motion described, with no mention of what caused it.
• Forces and Newton’s laws — what makes motion change.
• Work, energy and power — the bookkeeping of what a force costs.
• Momentum and collisions — what survives an impact.
• Circular motion and rotation — the same laws, rewritten for spin.
• Gravitation and oscillations — orbits, pendulums and springs.

Every Symbol Is a Quantity, and Every Quantity Has a Unit

A formula is only half the information. The other half is what each letter stands for and what unit it must be in — and that second half is where marks are actually lost.

Mechanics runs on three SI base units and a small set of derived ones built from them. The NIST guide to the SI sets out how those derived units are constructed; here are the four reference values that appear again and again below.

Notice that the newton and the joule are not new ideas — they are shorthand for combinations of kilograms, metres and seconds. That fact is what makes the unit check in the section below work at all. If you want the full picture of base and derived units, we cover it in the guide to SI units in physics.

The Mechanics Formulas Cheat Sheet: 40 Equations in 6 Families

Below are the 40 mechanics formulas that cover essentially every problem set at school and first-year university level. Each table gives the quantity you are solving for, the equation, every symbol with its SI unit, and a link to the calculator that does the arithmetic.

Scan the left-hand column first. Find the quantity you want, and the row hands you the formula.

1. Kinematics: Motion Without Forces

Kinematics describes motion using only position, velocity, acceleration and time — it never asks what caused the motion. The four constant-acceleration equations are the workhorses here.

Each of the four constant-acceleration equations leaves out exactly one of the five quantities. That is not decoration — it is the selection rule. Pick the equation missing the thing you neither know nor want, and we unpack why in the guide to the SUVAT equations.

2. Forces and Newton’s Laws

Force equations tell you how motion changes. Newton’s second law is the hinge that every other force formula swings on, and the rest of this family is really just different ways of working out what goes into F.

One detail worth pinning down: the F in F = ma is always the net force — the vector sum of everything acting. Draw the free-body diagram before you write the equation, a habit set out in full in Newton’s laws of motion.

3. Work, Energy and Power

Energy formulas let you skip the details of the journey. If a question gives you heights, speeds and distances but never mentions time, this family is almost always the shortcut.

The joule doing the work here is the same joule that heats a kettle. That single currency running across mechanics, heat and electricity is the point of the whole idea, explored further in what energy is in physics.

4. Momentum and Collisions

Momentum formulas govern what happens when things hit each other. Their power is that total momentum is conserved in every collision — even the messy ones where energy is clearly lost to noise, heat and crumpled metal.

A quick sanity check students find useful: N·s and kg·m/s are the same unit written two ways. If your impulse and your momentum change do not come out numerically equal, one of them has an arithmetic error.

5. Circular Motion and Rotation

Rotational formulas are the linear ones in disguise. Swap distance for angle, mass for moment of inertia and force for torque, and the shapes of the equations repeat almost line for line.

Angles must be in radians whenever they sit inside ω, α or a period. Feed a calculator degrees where it expects radians and the answer comes out wrong by a factor of about 57.

6. Gravitation and Oscillations

These formulas handle the two situations where motion repeats: orbits and oscillations. Both come down to a restoring influence pulling something back towards where it started.

How Do You Choose the Right Mechanics Formula?

Choose a mechanics formula by naming the quantity you want, listing the quantities you were given, and then picking the family whose equation contains only those symbols and nothing else. Three steps, in this order:

1. Name the target. Write down the symbol and unit of the one thing the question asks for — “v, in m/s”, not “the speed bit”.
2. List what you have. Extract every number with its symbol and unit. Do this before you look at any formula.
3. Match, don’t hunt. Pick the equation whose symbols are covered by your list plus your target. If one symbol is left over that you neither know nor want, that is the wrong equation.

Step three is where the sheet above earns its keep. Because the tables are keyed by “you want”, you are searching a short column rather than scanning forty equations.

Routing from what a question gives you to the right family of mechanics formulas.

How Do You Rearrange a Mechanics Formula?

Rearrange a mechanics formula by doing the same operation to both sides until the symbol you want stands alone — always in symbols first, before any number goes in. Substituting early is the single most common source of arithmetic slips.

For the many mechanics formulas shaped like A = B × C — F = ma, p = mv, W = Fd — there is a shortcut worth internalising. Cover the symbol you want: if it was on top you multiply the two below, and if it was on the bottom you divide the top by the other.

The lab below lets you try it. Pick a relationship, choose which variable to solve for, and watch both the rearranged equation and the unit check update as you move the sliders.

One warning the lab makes obvious: the triangle only works for the A = B × C shape. An equation like v² = u² + 2as has four terms and a square in it, so no triangle will save you — that one has to be rearranged properly, line by line.

How Do You Check a Mechanics Formula With Units?

Check a mechanics formula by replacing every symbol with its SI unit and simplifying: if the right-hand side does not collapse to the unit of the quantity you wanted, the formula is wrong. This takes about ten seconds and catches a whole class of errors that re-reading never will.

Take kinetic energy. Substituting units into ½mv² gives kg × (m/s)², which is kg·m²/s². Since 1 J = 1 N·m = 1 kg·m²/s², the units land exactly on the joule — the formula survives.

Now try the classic slip of writing kinetic energy as ½mv. That gives kg·m/s, which is momentum, not energy. The unit check flags it instantly, without you needing to remember which version was right.

Three habits make this reliable in practice:

• Convert to SI before substituting. Grams to kilograms, centimetres to metres, minutes to seconds — every time.
• Strip the pure numbers. Factors like ½, 2π and cosθ carry no units, so ignore them during the check.
• Expand the named units. Rewrite N as kg·m/s² and J as kg·m²/s², then cancel.

A unit check confirms a formula is dimensionally possible; it cannot catch a wrong numerical factor. Both ½mv² and 2mv² pass the check, so it is a filter, not a proof. For the wider method that surrounds this step, see our six-step framework for solving physics problems.

Where These Mechanics Formulas Show Up in Real Life

Every equation on this sheet was written to describe something someone needed to build or predict. Four examples where they are still doing exactly that:

• Crumple zones. Impulse says J = FΔt. A car body that deforms stretches Δt across a longer collision, so the same change in momentum arrives as a smaller average force on the occupants.
• Motorway curve design. The banking angle formula tanθ = v²/(rg) sets the camber engineers build into a bend so vehicles at the design speed need no help from friction to turn.
• Figure skating spins. Angular momentum L = Iω is conserved when a skater pulls their arms in. Moment of inertia drops, so angular velocity climbs — no push required.
• Satellite launches. Kepler’s third law fixes the altitude that gives a 24-hour period, which is why geostationary satellites all sit on the same ring above the equator.

Common Mistakes With Mechanics Formulas

The same four errors account for most lost marks in mechanics. Each one is a misunderstanding, not a slip — which is why they survive re-reading the sheet.

Mistake 1: Treating mass and weight as the same thing

Mass is measured in kilograms and never changes; weight is a force in newtons and changes with location. Writing 60 kg into a formula that wants a force in newtons is one of the most frequent errors on marked scripts.

The fix is mechanical: whenever you see a mass in a force equation, ask whether the equation wants m or mg.

Mistake 2: Using F = ma with only one of the forces

The F in Newton’s second law is the net force, not the force you happen to have been given. A 100 N push on a crate with 30 N of friction gives a net force of 70 N, and using 100 N inflates the acceleration by more than 40 per cent.

Mistake 3: Assuming kinetic energy is conserved in every collision

Momentum is conserved in all collisions; kinetic energy is conserved only in elastic ones. In a real crash a large fraction of the kinetic energy goes into deformation, heat and sound — so applying the elastic-collision equations to a car pile-up gives a confidently wrong answer.

Mistake 4: Mixing degrees and radians

Angles inside ω, α, or a period must be in radians; angles inside sinθ and cosθ in force resolution are usually in degrees. A pendulum or rotation answer that comes out about 57 times too large or too small almost always traces back to this.

How Mechanics Formulas Connect to the Rest of Physics

Mechanics formulas are not a closed set — they are the layer everything else is built on top of. The joule you calculate with ½mv² is the same joule that appears in specific heat capacity; the newton from F = ma is the same newton in electrostatic and magnetic force laws.

Within mechanics itself, the families are alternative routes rather than separate subjects. The diagram below shows the same block reaching the same answer two different ways.

Force and energy routes through the same mechanics problem, giving identical answers.

Add friction to that slope and the energy route needs an extra term, but it still works. Add a collision, and momentum becomes the only route that survives. Choosing well is what separates a two-line solution from a twenty-line one.

If you need equations from outside mechanics — thermodynamics, waves, electromagnetism or modern physics — they are all collected in the physics formulas hub. MIT’s introductory mechanics course makes the same argument this page does, and makes it forcefully: its own 8.01 Classical Mechanics materials open by contrasting learning a set of formulas to memorise with learning principles you can apply to unfamiliar situations.

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