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## Main Question or Discussion Point

I was reading through Lewis Epstein's book "Thinking Physics" the other day, and I came across the problem called "Rolling Drain." (This is not a homework problem! It is a question about the definitions of force, momentum, and acceleration.)

The problem is simple: a cart full of water is rolling horizontally along on a frictionless track, and it is dropping water out a hole in the bottom as it goes. The questions are what happens to the cart's speed, momentum, and kinetic energy. For each quantity, does it a) increase, b) decrease, or c) stay the same.

Now, the speed is not going to change, because there are no (horizontal) forces acting on the system from outside that would change the speed. However, the mass is decreasing, since the system is losing water. These together mean the momentum and kinetic energy must decrease, since mass decreases while speed stays the same. Right?

My trouble comes when I try to think of this problem in terms of forces. On the one hand, it seems the system cannot accelerate, because no outside forces act on it. On the other hand, the system's momentum is changing, and according to NL2, force is defined as a change in momentum! So there MUST be a force here, right?

Typically we first learn NL2 as F = m(dv/dt), since for most problems mass is constant. But for this problem, velocity is constant while mass changes: F = v(dm/dt). (Or at least I think; maybe I am wrong and this is my problem!) This is just as much a force as the first one, right? But I cannot think how to represent this force to myself in the context of the problem. Is it true that force as 'change in momentum' and force as 'cause of acceleration' are distinct? Should I not think of the quantity v(dm/dt) as a force at all?

Can anyone help to clarify all this for me? (I am quite familiar with the Lagrangian and Hamiltonian formulations of mechanical principles, so I would also welcome any light that could be shed from that perspective, also. But I doubt that will be necessary!)

The problem is simple: a cart full of water is rolling horizontally along on a frictionless track, and it is dropping water out a hole in the bottom as it goes. The questions are what happens to the cart's speed, momentum, and kinetic energy. For each quantity, does it a) increase, b) decrease, or c) stay the same.

Now, the speed is not going to change, because there are no (horizontal) forces acting on the system from outside that would change the speed. However, the mass is decreasing, since the system is losing water. These together mean the momentum and kinetic energy must decrease, since mass decreases while speed stays the same. Right?

My trouble comes when I try to think of this problem in terms of forces. On the one hand, it seems the system cannot accelerate, because no outside forces act on it. On the other hand, the system's momentum is changing, and according to NL2, force is defined as a change in momentum! So there MUST be a force here, right?

Typically we first learn NL2 as F = m(dv/dt), since for most problems mass is constant. But for this problem, velocity is constant while mass changes: F = v(dm/dt). (Or at least I think; maybe I am wrong and this is my problem!) This is just as much a force as the first one, right? But I cannot think how to represent this force to myself in the context of the problem. Is it true that force as 'change in momentum' and force as 'cause of acceleration' are distinct? Should I not think of the quantity v(dm/dt) as a force at all?

Can anyone help to clarify all this for me? (I am quite familiar with the Lagrangian and Hamiltonian formulations of mechanical principles, so I would also welcome any light that could be shed from that perspective, also. But I doubt that will be necessary!)