Newton’s Third Law Simulator

What Is the Newton's Third Law Simulator?

How to use the Newton’s third law simulator

1. Read the four cards before you touch anything. They are already answering for a light trolley pushing a heavier one: Pair force  F 40.00 N, Acceleration of A  a_A 20.00 m/s², Acceleration of B  a_B 5.000 m/s² and Ratio a_A : a_B  inverse mass ratio 4.000 : 1. The three sliders stand at 2.0 kg, 8.0 kg and 40 N.
2. Check which body each subscript belongs to before you read a sign. The first card’s sub-line puts the convention in front of you, the same on both bodies — F_12 = -F_21, and the lab takes the first subscript to be the body that feels the force. The Newton’s third law examples guide is where that habit of naming both bodies is argued out and applied to seven everyday cases; this panel only ever shows one.
3. Now drag Mass of B and nothing else, and keep your eye on the first card. Take it to 80.0 kg and the Ratio card reads 40.00 : 1; take it to 2,000.0 kg and it reads 1000 : 1. Through both of those the Pair force card has not moved off 40.00 N, and neither has either product row.
4. Then move the Pair force slider instead, and watch the opposite happen. From 40 N to 400 N on the trolleys, a_A goes 20.00 to 200.0 m/s² and a_B goes 5.000 to 50.00 m/s² — and the Ratio card sits there reading 4.000 : 1 in both states. One slider owns the ratio; the other owns the size.
5. Read the four stats rows under the cards, bottom two first. Sum over the pair is 0.000 N and Sum on A alone is 40.00 N, at the same instant, on the same setting. Those are different sums over different things, and the gap between them is the whole of the do-they-cancel question. The guide to common physics misconceptions takes that one apart with a figure.
6. Read the two sentence strips below the canvas, not just the cards. The first names both forces with both bodies attached; the second turns the one force into the two accelerations. The second one rewrites itself when the masses match: press Equal masses and it reports equal forces, and here equal masses, so the two accelerations come out equal as well, where every other preset reports the inverse ratio instead.
7. Watch the drawing for a few seconds. Two arrows leave the contact point in opposite directions, drawn to the same length whatever the masses are, and the bodies themselves slide apart in the ratio the fourth card reports. The drift is already running when the page loads; the button beside Reset reads Pause motion, so press it when you would rather have a still frame to read.
8. Press Reset to get back to a known state. It returns all three sliders to 2.0 kg, 8.0 kg and 40 N and leaves the drift running. Nothing on the panel is stored between visits, so a reset is the fastest way to undo a drag you have lost track of.

Dragging is the right way to find out which reading a control is allowed to move, and a poor way to hit an exact figure — the mass sliders step in tenths of a kilogram and the force slider in whole newtons. When you already know the pair force and want one acceleration typed out, the Newton’s second law calculator does that step on one body at a time. Run it twice, once per body; it has no second mass box and no ratio, which are the two things this panel adds.

Worked example: change one mass at a time

Nothing in the two tables below was worked out here. Each row is a setting the lab was put into and then read back from, cell by cell, so every string in them is the panel’s own. The first table moves exactly one slider between consecutive rows; the second holds the five preset cases side by side.

Rows one to four are the walk the page is built on. Only Mass of B moves, from 2.0 kg to 2,000.0 kg, and the Ratio card goes 1.000, 4.000, 40.00, 1000 : 1. The Pair force card reads 40.00 N in all four, and so do both product columns. A thousandfold change in one mass, and the force between the two bodies has not budged.

Rows five to seven do it the other way round and settle the point. Both masses are now pinned at the slider ends, 0.1 kg against 2,000.0 kg, and only the force moves: 40 N, then 2,000 N, then 1 N. Every acceleration in those three rows is different, and the Ratio card reads 2.000e+4 : 1 in all three. The force sets the size of both accelerations; the masses alone set how they divide.

Rows eight and nine are the same pair of masses exchanged. At 0.1 kg each the two accelerations are both 10.00 m/s²; swap 2,000.0 kg into A and the ratio inverts to 5.000e-5 : 1 while Sum over the pair stays at 0.000 N. The label on a body is a choice; the pair force is not.

The last three rows are the force slider swept with the trolleys left alone. At 1 N, 400 N and 2,000 N the two product columns read 1.000, 400.0 and 2000 N — equal to each other and to the card in every row. That is m_A a_A = m_B a_B = F read off the panel rather than derived, and it is the arithmetic the Newton’s second law calculator does one body at a time.

The five preset cases side by side

These are the only five settings that give the scene a title. The last column is the tail of the second sentence strip, and the top row has its own wording because the lab rewrites that sentence when the two masses match.

Read the two sum columns down the page. Sum over the pair is 0.000 N in all five cases and Sum on A alone is the full pair force in all five. The first column is why people think a pair cancels; the second is why nothing of the sort happens to A. Both are on screen at once, which is the only reason this panel can settle the question instead of asserting an answer.

Formula and symbol reference

The lab prints its own relations under the canvas, in the fixed definition line. Written out in the plain-digit form the panel uses, with every subscript as an underscore:

F_12 = -F_21, where F_12 is the force on body 1 exerted by body 2. Then a_A = F / m_A and a_B = F / m_B, the second law applied to each body on its own, from which a_A : a_B = m_B : m_A and m_A a_A = m_B a_B = F. No modulus bars appear anywhere on this page, because a bar either side of a symbol draws as a capital letter I in the site’s fonts.

The condition this panel cannot test for you is the one about force types. Both halves of a pair have to be the same kind of force, and the only kind on this track is contact between two touching bodies — there is no friction, no weight and no normal force to confuse it with. The guide to types of forces sorts those out, and the centrifugal force guide deals with the candidate that fails on a different condition altogether.

Ten rows follow and only three of them are something you drag. The rest are cards, stats rows or quantities the panel names without giving you a control for. For the sliders the right-hand column gives the end stops and the step; for everything else it gives the strings the lab prints and the span it prints them over.

If you read one row twice, make it the pair of stats rows at the bottom. They are the only two readings on the panel that are deliberately about each other, and a reader who has them side by side will not again be told that a third-law pair cancels out.

The physics: why the pair force ignores the masses

A pair force is a property of the interaction, and an interaction has one magnitude. Both bodies are in it, so both feel the whole of it; nothing is halved and nothing is shared. The panel makes that checkable rather than merely stating it: hold the force slider still, sweep either mass over its full four-decade range, and the first card never moves.

What the masses do change is what each body does with that force, and the second law decides that one body at a time — the guide to Newton’s laws of motion sets the three laws out together. Divide the two accelerations and the force cancels, leaving the masses the other way round; multiply each back and the one force reappears. Both of those are on the panel simultaneously — the fourth card and the two product rows — which is why no step of the argument has to be taken on trust here.

The two forces never cancel, and the reason is arithmetic rather than rhetoric. Cancelling is what forces do inside one body’s own sum, and these two are never in the same sum. Sum over the pair and Sum on A alone are the two sums printed side by side, reading 0.000 N and the whole pair force at the same instant. The free-body diagram guide is where choosing which body to sum over becomes a drawing habit.

This is also where the third law stops being a curiosity and starts being useful. Equal and opposite forces acting for the same time on two bodies change their momenta by equal and opposite amounts, which is why an isolated pair keeps the momentum it started with — the conservation of momentum guide works that through. This panel stops at forces and accelerations: it has no clock and reports no velocity.

Where the Newton’s third law readings break down

Every slider position this lab can reach is answered, and the arithmetic is not where it can let you down. The drawing is, and so is the number formatter. Seven limits sit under the readings on this page, and six of them are choices the lab makes on purpose rather than failures; the seventh is a boundary in the law itself rather than in this panel.

An arrow length is not a force reading, and the drawing never claims it is

The two arrows are equal to each other at every setting, and that equality is the physics. Their length grows with the square root of the pair force rather than in proportion to it, at 5.6 pixels per square root of a newton, with an 18 pixel floor and a ceiling set by the room the scene has left, because the slider spans 1 N to 2,000 N. On a 612 pixel canvas that gives arrows of 18.0, 35.4, 56.0, 112.0, 177.1 and 205.2 pixels at 1, 40, 100, 400, 1000 and 2,000 N. Read the force off the card.

Nor is a body’s drawn size a mass reading, and the scene says so when it has to

A radius grows with the cube root of the mass and is then held between a floor and a ceiling that scale with the canvas, 9 and 34 pixels at full width, so that a twenty-thousand to one ratio still shows two visible bodies instead of one dot and one wall. When that clamp ties the two radii while the masses differ, the scene tries to say so in words, and where the note will not fit the mass labels under the bodies carry the reading on their own. Those labels, and the chips above the sliders that agree with them, are the honest reading of a mass — never the drawn size.

A chip and a card print the same setting differently, and neither is wrong

Push the force slider to the top and the chip reads 2,000 N while the Pair force card reads 2000 N. The chip echoes a setting and takes a thousands separator from 1,000 upwards; the cards print computed figures at four significant figures, and 2000 already carries four. The same split shows on the masses, where the chip reads 2,000.0 kg and the ratio sub-line repeats it as = m_B : m_A = 2,000.0 : 2.0.

Four significant figures is the ceiling, and past it the readings switch to exponent form

Every input here is an exact value you chose on a slider, so there is no measurement uncertainty to propagate and nothing in the physics fixes the number of digits. The lab stops at four with the trailing zeros kept, which is why an acceleration reads 5.000 m/s² rather than 5. Outside that window the same formatter writes 2.000e+4 : 1 and 0.0005000 m/s², which are readings rather than overflows.

Zero mass and zero force are off the grid on purpose, so no reading is ever undefined

The mass sliders stop at 0.1 kg and the force slider at 1 N. That is what keeps dividing the force by a mass from being a division by zero, and keeps the ratio of the two accelerations from being zero over zero. There is no setting on this panel at which a card has to refuse, which is unusual for a lab here and is a consequence of where the sliders end rather than of anything clever in the arithmetic.

The track is deliberately bare, and three forces you might expect are simply absent

No gravity, no normal force and no friction are modelled in this panel: two bodies touch on a frictionless horizontal track and one contact pair acts between them. The track is drawn, but no force from it enters the arithmetic, which is why a weight and a normal force never appear on either body, and why the lab’s own fixed line says outright that those two are not a third-law pair. For which arrows belong on a single body at all, the free body diagram lab is the panel built for the question.

The pair statement itself has a boundary, and it is well outside anything here

As written, the pair statement covers two kinds of force: contact between touching bodies, and Newtonian gravity. Between them those account for every setting this panel can reach. Once two charges are moving past one another the field between them carries momentum of its own, which has to go on the books before the two forces balance, and what survives in general is the conservation of momentum rather than the force pair by itself. The Wikipedia article in the references below is the source for that boundary; no figure on this page depends on it.

Where Newton’s third law is actually used

Aerodynamics, where the whole subject leans on the pair

NASA’s Glenn Research Center sets out three cases on its beginner’s guide page on Newton’s laws of motion: a jet engine throwing gas backwards and being pushed forwards, a spinning ball deflecting air to one side and curving to the other, and a wing deflecting air downwards and being pushed up in return. Each is one interaction with two named bodies, which is the shape this panel draws; a wing is the one of the three with an argument of its own behind it.

Two people pushing off, which is the case the panel models most exactly

Press Two students push off: 50.0 kg against 80.0 kg with a 120.0 N pair force between them gives 2.400 and 1.500 m/s², a ratio of 1.600 : 1. Both of them really do accelerate, in opposite directions, from one push. On wheeled chairs there is almost nothing else acting horizontally, which is exactly the frictionless track this lab draws.

Rolling stock and trolleys, where the contact pair is the whole of the engineering

A coupling or a buffer carries one force with two ends, and the design question is what each end does to the vehicle behind it. Trolley pushes trolley is the arithmetic in miniature: 40.00 N on each of a 2.0 kg and an 8.0 kg body, giving 20.00 and 5.000 m/s². The light vehicle is the one that gets thrown about, and the force on it is not the reason why.

Anything a person pushes that is far heavier than they are

Child pushes a car is the everyday version: 30.0 kg and 1,200.0 kg with 150.0 N between them, giving 5.000 against 0.1250 m/s² and a ratio of 40.00 : 1. The car moves, and at a fortieth of the rate. This is the setting that makes the law look false in daily life, and the one where the Pair force card is most worth staring at.

Recoil and separation problems, once a clock is added

Give the pair a duration and the equal and opposite forces become equal and opposite changes of momentum, which is how a rocket, a gun or a pair of skaters is actually analysed. This panel has no clock, reports no velocity and does none of that: the conservation of momentum guide is where the time interval comes in. What the panel contributes is the step before it, with both bodies named.

Where to go next

A panel can show you which readings a control is allowed to move, and that is the end of what a panel is for. Newton’s Third Law: Examples is where the subject itself is written out: the three-condition pair test, seven everyday cases with both bodies named, the book on the table and the lift that separates its weight from the table’s push, the tug of war, the horse and cart, and eight worked problems. The calculator that finishes the arithmetic one body at a time heads the related tools below.

For the law in its place among the other two, the guide to Newton’s laws of motion is the overview, and the Newton’s laws of motion lab puts all three in one panel. The Newton’s second law lab is the single-body version of the step this one runs twice.

Nearer the questions this panel raises, the guide to common physics misconceptions is where the cancelling claim is taken apart, the free-body diagram guide is which arrows belong on one body, the guide to types of forces is the force-type condition, and the centrifugal force guide settles the candidate that fails on the different-bodies condition.

Further out, the conservation of momentum guide is what the pair becomes once a time interval is involved. Beyond that there is the simulation library to work through, an index of calculators for when typing beats dragging, and the blog for everything written out at length.