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?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.