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## Homework Statement

This is problem 3.9 from Kleppner & Kolenkow's Intro to Mech.

A freight car of mass M contains a mass of sand of mass m. At t = 0, a force is applied in the direction of motion, and simultaneously sand is let out of the bottom of the car at a rate dm/dt. Assuming the car is at rest at t = 0, find the speed of the freight car when all the sand is gone.

## Homework Equations

$$\vec{F} = \frac{\mathbb{d}\vec{p}}{dt}$$

$$\vec{p} = m\vec{v}$$

## The Attempt at a Solution

I attempted to form the differential of the momentum of the system, as follows:

$$p(t) = \big(M + m - (\frac{dm}{dt})(t)\big)v$$

$$p(t + dt) = \big(M + m - (\frac{dm}{dt})(t + dt)\big)\big(v + dv\big)$$

$$dp = p(t + dt) - p(t) = -\frac{dm}{dt}(v dt + t dv + dv dt) + M dv + m dv$$

Dividing through by dt and neglecting a leftover dv, I get:

$$\frac{dp}{dt} = \big(M + m - t \frac{dm}{dt}\big)\big(\frac{dv}{dt}\big) - v \frac{dm}{dt}$$

Unfortunately, I have very little idea of what to do from here. Furthermore, thinking about the problem more makes me realize that I have no real idea of how to utilize the force F. It seems reasonable to place it into my last equation, but that seems a bit arbitrary to me and I can't imagine how I would even progress from there. I'm sure I'm making this more complicated than it should be, but still, some help would be greatly appreciated. Thanks!