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## Homework Statement

Find the gain in kinetic energy when a rocket emits a small amount of matter. (Well, there's more, but the rest of the question is built around this initial problem)...

## The Attempt at a Solution

My logic revolves around the total kinetic energy of the rocket after ejecting matter, which I took to be the kinetic energy

*before*the emission added to the gain in kinetic energy. Furthermore, the mass of a rocket emitting a small amount of matter is reduced by dM = -dm and its velocity increases wth dv to (v+dv). If we let u = velocity at which matter is ejected, then after combining all of the above into a single equation we have:

[tex]K_{f}=K_{i}+K_{gain}[/tex]

[tex]\frac{1}{2}(M-dm)(v+dv)^2=\frac{1}{2}Mv^2+\frac{1}{2}dm(v-u)^2[/tex]

But this doesn't seem to be leading in the right direction and I personally believe there is a flaw in my reasoning.

So how am I supposed to set up this problem?

Thanks!

phyz