Need help picking vars to integrate on a neg-acceleration prob

[white 2.5rs]

This is extra credit, and the test has passed (no one got it), the prof said he'll still take efforts on it so here is my attempt.

A rocket sled is going down a test track at 180 km/hr (calculated to 50 m/s). It drops a snorkel into a trough of water. This diverts 30 kg/s of water vertically. The sled has mass of 500kg (including the snorkel and water in it).

The question is how far will the sled travel once it drops the snorkel.

I know I need to integrate from 50 to 0... but beyond that I don't know what vars to integrate in order to get something to plug into kinematic equations...

negate fluid+track+air friction.

 

[kuruman]

The statement of the problem is not very clear about the role of the snorkel. The only interpretation that is reasonable is that the snorkel sucks in the water vertically and then drops it out the top also vertically. As the problem states, the mass of the sled plus water is constant. The rocket sled is slowed down because it has to accelerate 30 kg of water per second from zero to whatever speed it has at any particular moment. Under those assumptios we can write Newton's second law for the sled plus water system as $$M\frac{dv}{dt}=-v\frac{dm}{dt}.$$ Here $M$ is the mass of sled + water (500 kg), symbol $v$ stands for the speed of the sled and $\frac{dm}{dt}=\mu$ is the rate at which water enters the sled (30 kg/s). Then,
$$M\frac{dv}{dt}=-\mu v~\rightarrow~\frac{dv}{v}=-\frac{\mu}{M} dt.$$The solution of this differential equation is
$$v(t)=v_0e^{-\frac{\mu}{M}t}$$
Clearly the sled will stop ($v=0$) when $t\rightarrow \infty$. The displacement of the sled is
$$\Delta x=\int_0^{\infty}v_0e^{-\frac{\mu}{M}t}=\frac{Mv_0}{\mu}$$
Answer (2 sig figs): $$\Delta x =\frac{500 ~\rm{kg} \times 50~\rm{m/s}}{30~\rm{kg/s}}=8.3\times 10^2~\rm{m}.$$
Note: You may not use the standard SUVAT kinematic equations because they are valid only under constant acceleration which is not the case here.