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In Morin’s Mechanics book, a problem called leaky bucket goes:

5.17. Leaky bucket ** At $t = 0$, a massless bucket contains a mass $M$ of sand. It is connected to a wall by a massless spring with constant tension $T$ (that is, independent of length).25 See Fig. 5.23. The ground is frictionless, and the initial distance to the wall is $L$. At later times, let $x$ be the distance from the wall, and let $m$ be the mass of sand in the bucket. The bucket is released, and on its way to the wall, it leaks sand at a rate $dm/dx = M/L$. In other words, the rate is constant with respect to distance, not time; and it ends up empty right when it reaches the wall. Note that $dx$ is negative, so $dm$ is also.

(a) What is the kinetic energy of the (sand in the) bucket, as a function of $x$? What is its maximum value?

(b) What is the magnitude of the momentum of the bucket, as a function of $x$? What is its maximum value?

In the solution, it starts:

The initial position is x = L. The given rate of leaking implies that the mass of the bucket at position x is m = M(x/L). Therefore, F = ma gives −T = (Mx/L)x ̈.

My question is, how can they use F=ma? Wouldn’t you need to know the speed that the mass is being ejected, and then use F = dp/dt? I understand that the solution must be correct, but I don’t get why F=ma is valid to use. I’d appreciate any explanation. Thanks!

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The sand is not being “ejected”. It’s just falling away. So at each instant you can say “there’s a bit of sand dm and the remaining sand M in the bucket, and from now on I’m just paying attention to the forces on M” leaving the little bit of leakage at that instant to do what it does.

F = ma is then the force (on the bucket) that’s acting on the mass (in the bucket) which nicely matches up all the parts.

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