What It Takes to Stop Something
1. Momentum is inertia in motion
Chapter 2 said that things resist changes in their motion, and called that inertia. It is a property of the object and it sits there whether the object moves or not. Momentum is the version of that idea for something already moving: how much motion it has, and therefore how much trouble it is going to be to stop.
Two things decide it, and they both count. A heavier thing has more momentum than a lighter one at the same speed. A faster thing has more than a slower one of the same mass. So a supertanker drifting slower than you walk and a bullet you cannot see are on the same list, at opposite ends of it, and one number has to carry both facts at once.
2. Changing it takes a force that lasts
To change something’s momentum you need a force. That much is Chapter 4. What this chapter adds is that the force has to last: a shove that ends immediately changes almost nothing, and a gentler push held for ten seconds can do a great deal.
The force together with the time it acts is called the impulse, and the impulse is what actually changes the momentum. This is why a follow-through is coached in every sport that involves hitting something. The club is not stronger for staying in contact. It is in contact for longer, and that is a bigger impulse.
3. Once the change is fixed, all you can choose is the time
Here is where the chapter turns useful. Suppose you are falling and about to arrive. The momentum you have to lose is already decided by how fast you are going and how much of you there is. Nothing about the landing can change that. The stop has to happen.
What is still open is how long the stop takes. Squeeze it into a thousandth of a second and the force is enormous. Stretch it over half a second and the same change is spread thin. Every piece of safety equipment ever invented is that one sentence in a different shape: airbag, crash barrier, bent knees, boxing glove, gymnast’s mat, the packaging around anything fragile.
The same fall, arriving on different things. The bar on the left is the momentum that has to be lost, and no setting of the slider changes it. What the slider changes is how long the stop is allowed to take, and the force follows from that. Drag it all the way down and you have built a brick wall.
Notice what the simulation will not let you do. There is no setting that reduces the left-hand bar. You cannot land with less momentum than you arrived with, so the only lever available is time, and every safety device in the world is pulling on it.
4. Bouncing asks for more
Stopping something takes impulse. Sending it back the way it came takes more, because you have to stop it first and then throw it. So anything that bounces off you hits you harder than the same thing arriving and staying put.
This is not a small effect. A ball that rebounds at close to its arrival speed delivers roughly twice the impulse of an identical lump of clay that hits and sticks. It is why hail dents a car roof that the same amount of rain leaves alone, why a water wheel with curved cups that turn the water right around beats a flat paddle, and why a boxer is taught to roll away from a punch rather than lean into it.
5. What a collision cannot change
Now add up the momentum of everything taking part, before and after. The remarkable thing, and it is the reason this chapter exists, is that the total is the same. Whatever went on in the middle, however violent, however messy, the books balance.
The reason is Chapter 5. The two objects push on each other with equal and opposite forces, for exactly the same length of time, so they receive equal and opposite impulses. Whatever one gains the other loses, and the total cannot move.
An explosion is the same rule read backwards. A cannon and its ball start with a total of nothing, so they must still add to nothing afterwards, which is why the cannon has to go the other way. A rifle into a shoulder, a rocket and its exhaust, and you stepping off a small boat are all the same sentence.
Two carts on a track. Set the masses and how fast the left one arrives, then choose whether they stick together or bounce apart, and run it. The two totals underneath are the momentum before and the momentum after. Nothing you can do to the sliders will make them differ.
Two settings are worth finding on purpose. Make the carts equal, set B at rest and choose sticking: they go off together at half the speed, which surprises almost everyone who expects either the full speed or nothing. Then make A very light, B very heavy and choose bouncing: A comes back nearly as fast as it arrived, and B barely moves. The total was unchanged in both cases, and it did not look remotely the same.
Check yourself
1. Rank these by momentum, most first, and say what you had to assume: a lorry parked outside; a bicycle ridden hard; a bullet in flight; a child sprinting.
The parked lorry is last, at nothing, and that is the point of the question: momentum needs motion, so its mass buys it nothing while it sits there. Above that, the child, and then the bicycle and the bullet in an order you cannot settle without numbers. A bullet is very fast and very light; a rider and bicycle are perhaps eighty kilograms at running speed. Naming the assumption honestly is worth more here than picking an order.
2. An egg thrown at a bedsheet survives and the same egg thrown at a brick wall does not. Say which quantity is the same in the two cases, which one is different, and which one breaks the shell.
The change in momentum is the same: both eggs arrive at the same speed and both end up stopped. The time is different, and so the force is different. The sheet does not reduce what has to happen, it only lets it happen slowly. The force is what breaks the shell. Watch for the answer that says the sheet reduces the momentum, which sounds right and is not.
3. A rubber ball and a lump of clay of the same mass hit a wall at the same speed. The ball bounces back, the clay sticks. Which one gives the wall the bigger push, and why?
The ball, by roughly a factor of two. The wall has to stop the clay, and it has to stop the ball and then send it back the other way, which is a larger change in momentum and so a larger impulse. Same mass and same arrival speed is what makes the comparison fair; the only difference left is what happened on the way out.
4. Two carts of equal mass, one rolling and one at rest, collide and stick together. Predict their speed compared with the rolling cart’s, and say why the answer is not zero.
Half. All the momentum is still there and it is now being carried by twice as much cart, so the speed halves. Zero is the popular wrong answer, and it comes from imagining that hitting something at rest uses the motion up. Nothing uses momentum up; it only moves around.
5. A student says momentum cannot be conserved in a head-on crash, because both cars end up stopped and stopped is nothing. Name what they have left out, and say where the momentum went.
If the two cars were identical and arrived at the same speed in opposite directions, the total was already zero before the crash, so ending at rest is exactly what conservation predicts. If they were not identical, the wreck does not end up at rest; it slides, and then friction with the road hands the rest to the Earth, which is far too massive for anyone to notice. Either way the thing left out is the system boundary.
Chapter 7, Energy. This chapter found one quantity a collision cannot change. The next one finds another, and that one can be followed all the way through a change and out the other side, into places momentum never goes.