Collisions
Momentum Always Survives. Energy of Motion Usually Does Not.
What No Collision Can Change
A cue ball strikes a pool ball. A hammer hits a nail. Two cars collide at an intersection. What happens at the moment of contact is a mess: the surfaces squash, ring, warm up and sometimes break. Yet one thing about every collision is simple, and it follows from the third law of motion. While the blow lasts, each pushes the other exactly as hard, for exactly as long. Whatever momentum one gains, the other loses. Add up the momenta, minding their directions, and the total afterward is the same as before.
There is another way to see the same rule. Every pair of objects has a balance point, the average position of all its mass, called its center of mass. It lies on the line between the two, nearer the heavier one. Out in empty space, where the two push only on each other, the pair as a whole obeys the first law. Its center of mass coasts on in a straight line at a steady speed, before the crash, during it and after it. On the ground the road and the air push on the pair too, but for the instant of a crash they are weak beside the blow, and the rule holds through it.
That is the rule’s power: it settles part of how a collision must come out without asking how it happens. A car runs into an identical parked car, and the two lock together. Whatever the metal does, the wreck moves off at half the speed, twice the mass carrying the same momentum. No exception has ever been found, from billiard balls to protons in the Large Hadron Collider. The reason runs deep. It follows from the laws of physics being the same in every place.
What a Collision Can Lose
Momentum is one way to measure motion. There is a second, and for half a century the two were argued over as if only one could be right. When hard balls collide, two totals come out unchanged: mass times velocity, minding directions, and mass times speed squared. One side argued that the second was the true measure of motion, because it measured what motion can do. Drop brass balls into soft clay, and a ball arriving twice as fast digs the same hollow as one four times as heavy.
Both sides were right, about different things. The second quantity is now called kinetic energy, the energy of motion: half the mass times the speed squared. It has no direction and is never negative, so it never cancels. Two identical cars meeting head-on at the same speed have no momentum in total, but a great deal of energy. And it grows with the square of the speed, so twice as fast carries four times the energy.
Unlike momentum, the energy of motion is often not kept. When the car locks onto the parked one, the wreck has twice the mass at half the speed: the same momentum, but only half the energy of motion. The other half went into the crash. A collision can only use up the motion of the two toward each other. It can never touch the drift of their balance point, so that much energy has to stay as motion. Two identical cars meeting head-on at the same speed have a balance point that stands still, so all of their energy of motion can go, and when they lock together, it does.
At the other extreme, nothing is lost. Hard steel balls come close, and Newton’s cradle shows what happens when a collision keeps both totals.
Why does one ball in send exactly one ball out? Two balls leaving at half the speed would carry the same momentum, but only half the energy, so that ending is ruled out. The two totals alone do not settle it, though: with three or more balls moving, they allow other endings. What settles it is that two equal balls meeting head-on, in a knock that keeps both totals, simply swap speeds. The speed is handed down the row from ball to ball, and real steel balls, even touching, come close to that. They also lose a little in every knock, which is why a cradle slowly winds down. Most collisions sit somewhere between the two extremes. A superball comes back at about nine tenths of the speed it arrived with, a tennis ball at about three quarters, and a lump of wet clay not at all.
Where the Energy Goes
The lost energy is not destroyed. It is broken up and spread among the tiny parts of the two bodies, as when someone changes a large coin into small change. The blow sets the atoms of both bodies shaking a little harder, each in a direction of its own, and that shaking is what we feel as warmth. Some of the energy leaves as sound, and some goes into bending metal that stays bent, most of which ends up as the same jiggle too.
This is why one total survives every collision and the other often does not. The shaking runs every way at once. Add up the atoms’ momenta and they cancel to nothing. Add up their energies and nothing cancels: each shake adds energy, whichever way it goes. So the shaking can hold any amount of energy and no momentum at all. Energy of motion can hide in warmth. Momentum has nowhere to hide.
The change runs only one way. Shaking atoms never fall back into step and send a wreck rolling off again. Nothing forbids it, but there are so vastly more ways for them to shake than to march in step that it never happens. That one-way street is the second law of thermodynamics, met here in its plainest form.
Atoms themselves can do what no ball can. Two argon atoms that meet in the air bounce apart with no loss at all. A ball loses energy by setting its atoms shaking, but an atom’s only parts are its electrons and its nucleus. Each can take energy only by changing to another of its fixed patterns, and in argon the cheapest change, in the electrons’ cloud, costs hundreds of times more than such a collision carries. With no way to take a little, the atom takes nothing. Molecules of air can also set each other spinning, but the spin is handed back in later collisions, so nothing drains away.
The Crash Is Settled Before It Starts
In a car crash, what hurts the people inside is how suddenly their speed changes. How much it changes is set almost entirely by momentum, before anybody designs anything.
Which is worse: driving into a solid wall at 50 kilometers an hour, or meeting an identical car head-on with both doing 50? The usual answer is the car, because the speeds seem to add up to 100. They do not, for the purpose that matters. The two cars carry equal and opposite momentum, so the total is zero, and their balance point stands still through the whole crash. The plane where they meet never moves. Each car is brought to rest from 50 against a surface that stays put, which is exactly what a wall does. The two crashes are the same.
Seen from a camera moving beside one car at a steady 50, the other car does arrive at 100. But the wreck then drops behind the camera at 50, so each car’s speed changes by 50, not 100. Driving into the wall at 100 really is far worse: twice the change of speed, and four times the energy to be turned into warmth and bent metal.
When the cars differ, momentum shares out the change of speed unevenly. Say a car meets a sport utility vehicle twice its weight head-on, both at 50. Their balance point moves at about 17 in the bigger vehicle’s direction. If the two lock together, the big one slows by 33, while the car goes from 50 forward to 17 backward, a change of 67. Bounce or not, the lighter car’s speed changes twice as much. It is two skaters pushing apart, run backward, and it is why the people in the lighter vehicle usually come off worse.
What a Crumple Zone Buys
The change of speed is settled the moment the cars touch. How long it takes is not. A force is the rate at which momentum changes, so the same change spread over ten times as long needs a tenth of the push.
Stop from 50 kilometers an hour in two centimeters of rigid steel, and the whole change happens in three thousandths of a second, with a push of about 490 times your own weight. Stop in seventy centimeters of metal built to fold, and the same change takes a tenth of a second, at about fourteen times your weight. That is what the front of a car is for. The seat belt and the airbag do the same for your body inside the car: they give it room to stop.
The usual explanation is that a crumple zone absorbs the energy of the crash. That is true, and it is the other half of the old argument. A push acting for a time changes momentum; a push acting over a distance changes energy. The folding front makes the stop longer in both, so the same momentum and the same energy are taken from you more gently. Neither total was going to be any smaller. Only the stop was made longer.
Those numbers assume a steady push all the way, the gentlest stop there is. A real crash pulse rises and falls and peaks well above its average, so they are a floor, not a forecast. How much a body can take depends on how long the push lasts and how widely it is spread. One racing driver survived a crash in which his car recorded 214 times the pull of gravity, for an instant.
Reading a Collision Backward
Because both totals are counted so exactly, a collision can be read backward: from what comes out, you can tell what was there. Physicists fired alpha particles, tiny charged bullets thrown out by radium, at metal. A thin gold foil let nearly all of them straight through, nudged by a degree or so. But some came back out of the side they had gone in: from a plate of platinum thick enough to stop the rest, about one in eight thousand. It was as if a shell fired at tissue paper had come back and hit you.
Collision arithmetic says what that means. A light thing can be thrown back only by something heavier than itself. The atom’s electrons are about seven thousand times lighter than an alpha particle, and could turn one by less than a hundredth of a degree. The turn also had to happen in a single encounter, which takes a push far stronger than charge spread through a whole atom could give. So nearly all of an atom’s mass, and a concentrated electric charge, sit in a tiny core at its center: the nucleus, tens of thousands of times smaller across than the atom. The rest of the atom is not empty: its electrons fill it as a cloud, far too light to turn an alpha particle back.
Missing momentum is read the same way. In one kind of radioactive decay the electrons came out with less energy than the decay released, so physicists proposed that an unseen particle was carrying off the rest. Later, measurements of the recoiling atoms showed it carried momentum too: the electron and the recoil did not balance. The particle, the neutrino, was caught decades after it was proposed. The W particle, which carries the force behind that same kind of decay, was found in collisions where a fast electron shot out sideways with nothing visible on the opposite side to balance it. Detectors at the Large Hadron Collider hunt for invisible new particles in just this way, by the momentum that goes missing across the beam.
Every particle collider is that gold foil with the numbers turned up. At the highest energies, the energy of motion a collision uses up can turn into new particles, with antimatter matching any new matter. And the reading is still the same two totals, momentum and energy, counted before and counted after, with each particle’s mass counted as energy.
