# Simple frame of reference problem (conservation of momentum?)

I'm having trouble wrapping my head around a particular concept. Suppose we have a machine that fires balls of speed $u$ at some mass rate $\sigma$ (of units $\frac{kg}{s}$) directly at a car of mass $M$. The car is initially at rest on a frictionless plane, but as you can see it will eventually move.

I simply wish to know how $\sigma$ relates to the mass rate of the car getting hit. I know that the velocity of the ball in this reference for example is $u-v_{car}$, so $\sigma'$, the mass rate that hits the car, is related to the given mass rate by $$\sigma' = \sigma\frac{(u-v_{car})}{u}.$$

I don't see why this is true. I feel that perhaps this came from the conservation of momentum of some small mass of balls, which i'm certain is $-2(u-v)dm$ because I have assumed an elastic collision. How can I come to this conclusion?

## 1 Answer

Imaging the balls on a string. You are launching N balls per second, at a velocity $u$. This means the distance between the balls is $u/N$. And $N$ balls per second will pass a certain point in space.

Now if the car is moving at a velocity $v$ (same direction as $u$), fewer balls per second can hit it - because subsequent balls on the string have further to travel. In fact the speed with which the balls approach the car is $u-v$, and since their distance is still $u/N$, the rate at which they arrive is $\frac{u-v}{u}N$. Which leads to your result.