Where the unit starts

Linear Momentum

Momentum is mass in motion, p = mv, and it is a vector. Almost all of the trouble in this unit comes out of that second half: momenta add as arrows rather than as sizes, so two large momenta can add to nothing, and a momentum reported without a direction is only half of an answer.

1. Mass in motion, with a direction attached

Unit 3 gave you one way to say how much motion a thing carries. Kinetic energy is a single number, it has no direction, and it grows as the square of the speed. Unit 4 gives you the other way. The linear momentum of an object is its mass multiplied by its velocity, p = mv, and it grows as the first power of the speed.

The unit has no special name. It is kg m/s, kilogram meters per second, and it is read exactly as it is written. If a quantity you have worked out does not come out in kilogram meters per second, whatever it is, it is not a momentum.

Momentum is a vector. It points the same way the velocity points, because mass is a positive number and multiplying a vector by a positive number stretches it without turning it. On a straight track, with right chosen as positive, a cart moving right has a positive momentum and a cart moving left has a negative one. That sign is part of the answer, not decoration on it.

2. The same speed is not the same momentum

Momentum answers a question that speed by itself cannot: how much motion is there to get rid of. A loaded shopping cart and an empty one rolling side by side at the same speed are not equally easy to bring to a halt, and the arithmetic says so before your arms do.

Both factors count once. Double the mass at the same velocity and the momentum doubles. Keep the mass and double the velocity and the momentum doubles again. That is where momentum parts company with kinetic energy, which doubles for the first change and quadruples for the second.

So a 4.0 kg object at 2.0 m/s and a 2.0 kg object at 4.0 m/s carry the same momentum, 8.0 kg m/s, while their kinetic energies are 8.0 J and 16 J. Equal momentum does not mean equal anything else.

Two things are worth noticing while you move the slider. The arrows match while the speeds do not, and the lighter cart has to be going faster to get there. And once they match, nothing about the two carts is the same except the product mv, which is the only thing momentum ever asks about.

3. A system has one momentum, and it is a sum

Draw a boundary around any group of objects and call it a system. The momentum of that system is the vector sum of the momenta of its parts. There is nothing else to the definition, and there is no separate formula to learn: you find each mv, you keep its sign, and you add.

Adding the sizes instead is the most common way to lose this. Two carts carrying 6.0 kg m/s each, moving toward each other, give a system momentum of 0, not 12 kg m/s. The sizes do add to twelve. The sizes are not what the question asked for.

Switch between the two boundaries without touching the slider. Neither object has changed, and the total has, because the question changed. Then look for the slider setting where the total for both objects together falls to zero, and notice that the two objects are still moving as fast as they ever were.

4. Collisions and explosions, where momentum earns its keep

A collision is an interaction in which the forces the objects put on each other are much larger than any net force reaching them from outside. For the fraction of a second two carts are in contact, the push between them is far bigger than the friction of the track, which is what allows you to set the outside world aside for the length of the interaction.

Because only the state just before and the state just after are examined, each object can be treated as a single point carrying a mass. Its shape, its spin, and where on it the contact happened never enter the work. That is permitted here for one reason: nothing inside the interaction is ever looked at.

An explosion is the same idea pointed the other way. Forces inside the system drive its parts apart: a compressed spring released between two carts, a person stepping off a skateboard. Nothing is handed in from outside, so whatever momentum the system had before, it still has after. A 4.0 kg object at rest that comes apart into a 1.0 kg piece at 6.0 m/s and a 3.0 kg piece must send the heavier piece the other way at 2.0 m/s, because (1.0)(6.0) = (3.0)(2.0).

The two momenta in that last example are equal in size and opposite in direction, and they are not free to be anything else. Topics 4.2 and 4.3 are about why.

5. Three ways this goes wrong

Adding magnitudes instead of vectors. Two momenta of equal size pointing opposite ways total zero. Write each one with its sign before you add anything, and the error cannot happen.

Reporting a momentum with no direction. A momentum of 2.0 kg m/s is an unfinished answer. East, west, up the ramp, or a sign attached to a stated positive direction: pick one and say it every time.

Treating momentum and kinetic energy as the same news. They disagree on purpose. Equal momenta usually mean unequal kinetic energies, and a total momentum of zero tells you nothing at all about how much kinetic energy a system holds.

For this course the arithmetic stays small. Two objects, sometimes three, on a straight line, with every velocity measured from the same chosen positive direction.

Check yourself