This is not a homework problem, just a question from an old exam I'm looking at. You have an elephant (M1) hanging on a sling, connected by a string and two pulleys to a hanging bucket of water (M2). The string doesn't stretch, so if you pull down on either the elephant or the bucket the other will move up. The elephant drinks water from the bucket at a known rate, so it's constantly gaining mass while the bucket is constantly losing mass. The problem is to find the velocity as a function of time.
I have a copy of the solution, but I don't fully understand the first step. They write:
d/dt(P1) = T - M1g - 2vd/dt(M2)
d/dt(P2) = T - M2g + 2vd/dt(M2)
I don't fully understand where they got the last term in each equation. Obviously it's due to the momentum transfer of the water as it's drunk. But I would have expected that to come in the following form:
T-M1g = d/dt(P1) = M1dv/dt + vd/dt(M1)
I.e. the second term is the momentum change due to varying mass. That's how I've always seen it done in the standard rocket problem, for example. But the solutions say that IN ADDITION to that term there is a force due to the mass transfer.
Can anyone explain where the momentum transfer comes from? Is there a general equation that derives this? Is dp/dt = sum of forces still correct in this case?