- #1

Haorong Wu

- 413

- 89

- Homework Statement
- A cart full of sand starts to move by a total force ##f##, which is parallel to the cart movement direction. The sand is lost at a rate of ##\rho## per second. What is the velocity of the cart?

- Relevant Equations
- ##fdt=d(mv)=mdv+vdm##

The solution is from ##fdt=d(mv)=mdv+vdm## and separate the variables and then integrate them.

But at first I tried this method. At time ##t##, suppose the mass of the cart is ##m##, and its velocity is ##v##. And suppose at time ##t+dt##, its mass will be ##m-\rho dt##, and its velocity becomes ##v+dv##. Now I use the impulse-momentum theorem, I have $$fdt=(m-\rho dt)(v+dv)+\rho dt v-mv=mdv.$$ This is clearly wrong, but I could not figure out the mistakes.

But at first I tried this method. At time ##t##, suppose the mass of the cart is ##m##, and its velocity is ##v##. And suppose at time ##t+dt##, its mass will be ##m-\rho dt##, and its velocity becomes ##v+dv##. Now I use the impulse-momentum theorem, I have $$fdt=(m-\rho dt)(v+dv)+\rho dt v-mv=mdv.$$ This is clearly wrong, but I could not figure out the mistakes.