A fast ball is hard to stop, and so is a heavy one. How hard something is to stop depends on both its speed and its mass. The product of the two is called momentum, written .
It is a quantity with direction. A ball running right and a ball running left differ in sign. Everything here happens along one straight line, so once rightward is taken as positive, the sign of a number is all that is needed.
The same can be made up in many ways. A light fast ball and a heavy slow one may carry the same momentum. The figure exchanges mass for speed, and the bar does not change its length.
What follows is how this changes, and when it does not change at all.
The starting point is . Acceleration is how the velocity changes, so for a constant force it can be written .
Put that in and multiply both sides by . This gives , and the left side is , the change in momentum itself.
The on the right is called the impulse: the force multiplied by the time over which it acted.
Together, . The statement has become that momentum changes by as much impulse as is delivered. The content is still the equation of motion, but in situations like a collision, where a strong force acts for a very short time, this form is far easier to work with.
In a real collision the force is not constant. It grows from the instant of contact, peaks where the bodies are most compressed, and falls away as they separate.
This causes no trouble. Treat the force as a function of time and draw an - graph, and the impulse is the area beneath it. A rectangle for a constant force, the area under a curve for a varying one.
The figure lays a rectangle and a hump of equal area over one another. The shapes differ, but with equal areas the change in momentum is the same.
A strong force over a short time or a weak force over a long one, the result is unchanged so long as the area is the same. The next scene is that reading.
Read the other way round, in terms of : .
When the momentum to be absorbed is fixed, stretching the time lets a smaller force do the job. All that happens is that the denominator grows.
Rolling with a fall, bending the knees on landing, putting an airbag in a car. Every one of them does the same thing: it lengthens the time taken to stop and lowers the force on the body.
The figure stretches the hump sideways while keeping its area, and the height comes down. The area is fixed by the change in momentum and cannot be moved. The shape is what can.