Module 3 · straight-line motion and Newton's laws
The three equations of motion are one fact about a graph — you will see that in ten minutes. Newton's three laws are not one fact. They are three separate claims that people often run together, and this module is about pulling them apart: the first law says there is a point of view from which the other two are true at all; the second says how much a force bends a motion; the third says forces always come in pairs. Along the way you will notice that "F = ma" is far stranger than it looks.
Draw a graph of velocity against time for something with constant acceleration. It is a straight line: it starts at u and climbs with slope a. On such a graph the area under the line is the distance travelled — that is the whole input; everything else is reading the picture.
A car goes from 10 m/s to 30 m/s with steady acceleration over 10 s. How far does it travel in those 10 s?
Do it in your head from the graph, not from a formula.
The three equations are three ways of describing that shaded region.
Slice the graph into thin vertical strips of width Δt. Over one strip the velocity is almost constant, so the distance covered is v·Δt — a strip's area. Add all the strips: total distance is total area. Make the strips thinner and the "almost" goes away. That idea — slice, add, then let the slices shrink — is what integration is, and you have just used it: s = ∫v dt. Every formula in kinematics — the part of physics that describes motion without asking what causes it — is this argument with a particular shape of graph.
They assume a is constant. A car with its foot to the floor does not accelerate uniformly; a falling body in air does not; a rocket burning fuel gets lighter and accelerates harder. In all of these the v–t line curves, the triangle becomes a curved region, and you either integrate properly or slice it numerically — which is exactly what Feynman does by hand in §9-6 of Chapter 9 below, and what Bench 3 of Module 6 does to build an orbit out of nothing but repeated small steps. The three equations are a special case that happens to cover most exam questions. Do not mistake the exam for the world.
For two thousand years the obvious truth was Aristotle's: things move only while something pushes them, and stop when the push stops. Look at any cart on any road and it is plainly correct. Galileo's genius was to notice that the obvious truth was a side-effect of friction — that things stop not because motion runs out, but because something is rubbing it away. He did two things about it. First, real ramps at 20°, 10°, 5°, each polished as smooth as he could make them. Second, and this is the part nobody had done before, he asked what the pattern was heading towards at 0° — an experiment in a world without friction at all, which nobody can perform.
Roll a ball down one ramp and up a second. On smooth ramps it climbs back to nearly the height it started from, however steep or shallow the second ramp is. Now lower the second ramp.
If the second ramp is made perfectly flat and perfectly smooth, the ball:
The simulation treats the ball as sliding, not rolling. A ball that rolls without slipping puts two-sevenths of its energy into spin, so it accelerates at (5/7)g sin θ, where θ is the ramp angle, and arrives at 5.9 m/s rather than 7.0 m/s. The heights are unaffected, which is why Galileo's argument survives; Module 8 does the rolling case properly.
Galileo's argument is a limit — a claim about where a pattern is heading, read off from cases you can do about one you cannot. At 20° the ball travels a certain distance along the second ramp to regain its height. At 10° it must travel further. At 5° further still. At 0° it can never regain its height, so it must travel forever. The experiment is impossible to do — there is no perfectly smooth ramp — and that is exactly why Aristotle never did it. Galileo did it in his head, and physics began.
The "tries to regain its height" shorthand in the bench caption is safe here and nowhere else. It stands in for the energy trade you will prove properly in Module 5 — height goes down, speed goes up, and back — but only when the ramp is smooth. Turn the friction slider up and watch the shorthand fail: some of the speed rubs away into warmth, the ball never gets its height back, and Aristotle looks right again.
The first law is not a special case of the second with F = 0. It is the statement that there exist frames of reference — call them inertial frames — in which the second law is true at all. Without it, you would not know which observer's "a" to put into F = ma. And there is not just one such frame: any frame moving at constant velocity relative to an inertial frame is inertial too, and the laws take the same form in all of them — you met the idea in Module 2 as the moving train that changed the numbers but not the collision.the point Feynman makes in Chapter 9 and Lewin in Lecture 6
That last point is the deep one. Sit on a bus that is braking. A ball on the floor rolls forward with nobody touching it: in your frame the first law is false. So the first law has a job to do, and it is not the job people think. It warns you that the braking bus is the wrong place to stand — the other laws do not work here — and it promises that somewhere there is a right place to stand, the road for instance, where they do. Module 6 is about what to do when you insist on staying in the bus — the section there on "centrifugal force" is this same problem, seen from inside a turning car.
Lewin clips a small electric fan to a cart on an air track — a hollow rail with holes along its top, blowing air upward, so the cart floats on a cushion of air and rubs against nothing. That is what makes the measurement clean: with friction gone, the fan's push is the only horizontal force there is. The fan pushes air backward, the air pushes the fan forward, and the cart accelerates with a steady force that Lewin can vary by changing the fan's speed. This is the cleanest way to measure Newton's second law rather than assume it — and measuring it is the point, because as Feynman insists, it is not a definition.
Two straight lines through the origin. Acceleration is proportional to force and to one-over-mass; the constant of proportionality is 1 because the newton is defined to make it 1 — one newton is the force that gives one kilogram one metre per second squared. That is the content of
Here is a trap that catches most students, and Feynman spends a whole chapter (12) on it. "Force is whatever produces acceleration. So F = ma just defines force. So it is empty." If that were true, the law would tell you nothing about the world — and it would be impossible for it to be wrong.
The escape is that forces have independent descriptions. A stretched spring pulls with a force that depends only on how far it is stretched. Gravity pulls with a force that depends only on masses and distance. Friction has its own rule. Each of these is a separate fact discovered by separate experiments. The second law is then a genuine claim: the acceleration you measure will equal the sum of those independently-known forces, divided by the mass. That could easily have been false. It is a triumph that it is true, and the fan cart is where you check it.
The lift you rode this morning: stand on bathroom scales inside it. Accelerating upward, the scale reads more than your weight. Two forces act on you — gravity mg down, floor N up — and their sum must equal ma, pointing up. So N − mg = ma, giving N = m(g + a). The scale reads N, not your weight, so it reads more, by exactly ma. Accelerating downward, a is negative and it reads less. If the cable snapped, the scale would read zero, and you and it would be in free fall together: the "weightlessness" of orbit is not the absence of gravity but the absence of a floor pushing back. Astronauts on the space station are falling toward Earth continuously; they just keep missing.
Newton's third law: whenever A pushes or pulls B, B pushes or pulls A back, just as hard, in the opposite direction. Not almost as hard, and not only when things are standing still. You cannot touch without being touched.
A horse is harnessed to a cart. The horse pulls the cart forward; by the third law, the cart pulls the horse backward, exactly as hard. Two forces, equal and opposite — and it is tempting to conclude that they must cancel, so the horse and cart can never get going at all. Carts plainly do move. Something in that sentence is wrong.
So why does the cart move?
The rope pulls the cart forward with a force T, and it pulls the horse backward with the same T. Why the same? Because the rope itself is light: a force of T at one end and anything other than T at the other would give a nearly massless rope an enormous acceleration. So one tension T does two different jobs — a help to the cart, a hindrance to the horse. Note that these two forces are not a third-law pair: they are both forces exerted by the rope, on two different bodies. The genuine third-law pairs here are rope-on-cart with cart-on-rope, and rope-on-horse with horse-on-rope. But you never add those two together. Here is the rule that makes the sum honest: to find how the cart moves you draw a box round the cart alone and sum only what acts on it; likewise for the horse. That box is a free-body diagram, and Bench 4 draws you three of them (cart, horse, and the pair together). A third-law pair, by definition, has its two forces in two different boxes — so third-law pairs cannot cancel, any more than a debt of yours can cancel a debt of mine. The horse's real engine is the ground: it pushes backward on the earth with its hooves, and the earth pushes it forward. Take away the friction under the hooves — ice — and the strongest horse in the world goes nowhere.
Write Newton's second law once for each box. Let T be the rope tension, f the friction dragging on the cart, P the ground's forward push on the horse's hooves, and mc, mh the two masses. The rope does not stretch, so horse and cart must speed up together — one shared acceleration a for both.
A third-law pair always has the form "A pushes B, B pushes A". The two forces are the same kind (both contact, both gravitational, both tension), the same size, opposite in direction — and on different bodies. If you ever find yourself with two "equal and opposite" forces on the same body, they are not a third-law pair; they are two unrelated forces that happen to be balancing at this moment — like the floor pushing you up and gravity pulling you down — and they will stop being equal the instant you accelerate. The floor's push and your weight are equal here only because you are not accelerating right now. The third-law partner of your weight is your gravitational pull on the Earth, equal in size to your weight, and (as Bench 3 of Module 2 showed) enough to move the planet.
A rocket in space has nothing to push against. It does not need anything. It throws mass backward, hard; the mass pushes the rocket forward with equal force. That is the third law with the "ground" replaced by the rocket's own exhaust — and vacuum is better, not worse, because in air the atmosphere presses back on the escaping gas and on the mouth of the nozzle, and every newton of that is a newton subtracted from the thrust. Every textbook that says "a rocket pushes against the air" is wrong, and so was the New York Times when it said so in 1920. It printed a correction in 1969, the day after Apollo 11 launched.
1. A stone is thrown straight up. At the top of its flight, its velocity is zero. Its acceleration is:
2. A 1000 kg car and a 10 kg bicycle collide head-on. Which experiences the larger force during the collision?
3. A parachutist has reached terminal velocity, falling at a steady 55 m/s. The net force on her is: