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CAF123
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Homework Statement
An open topped truck of mass ##M## is moving along a straight frictionless track at constant velocity ##v_o##. At ##t=0,## the truck enters a rain shower and starts to collect rainwater. The rain falls vertically. Consequently, the trucks mass increases at a rate of ##\lambda## per second.
1) Show that its velocity ##v(t)## at time t after entering the shower is $$v(t) = \frac{Mv_o}{M + \lambda t}.$$
2)If the truck had been fitted with a drain hole to prevent rain from accumulating, find its velocity at time ##t##, assuming that the water leaves the truck at zero relative velocity.
3) Compute the momentum of the system (the system being the truck plus water in 1) and truck in 2) as a function of time, and explain the qualitative difference between cases 1) and 2)).
Homework Equations
Equations will be derived from Momentum conservation
The Attempt at a Solution
1)Straightforward, but I have two questions. Momentum before entering rainstorm: ##Mv_o.## Momentum afterwards: $$M(t + δt) v(t + δt) = [M(t) + δ\dot{M}(t)][v(t) + δ\dot{v}(t)],$$ by Taylor expansion. Multiply out, ignore squared term, collect terms and simplify.
Two questions though: I took ##\dot{v}(t) = 0## here to attain the required 'show that'. Initially it was because I thought the truck is moving at constant speed, but now that I think about it , it is the velocity at time t we are interested in, so why does setting it to zero work? Also, I presume momentum is conserved in this case since if the mass increases, then the velocity of the truck must decrease.
2) and 3) I am struggling to start. I thought 2) might be similar to the decline in mass problem (ie. rocket eqn derivation), but then again the truck does not accumulate mass in the first place with the drain hole so I don't know if this is applicable.
Thanks for any advice