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In class today, my professor was teaching conservation of momentum. One example she used was an open cart rolling on a frictionless track while in the rain. As the cart collects water, the mass increases; and due to conservation of momentum the velocity must decrease, in order to keep momentum constant. However, the way I understand it, since velocity is decreasing, that means it must have negative acceleration, implying net force is no longer 0, implying momentum is not constant. What am I misunderstanding here?

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This is really the same situation as an inelastic collision between two objects, the cart and the water. Before the "collision", the cart has mass $M$ and horizontal velocity $v$, and the falling rain has mass $m$ and horizontal velocity $0$. The horizontal momentum of the system is therefore $P = Mv$.

After the "collision", the cart and the rain have the same velocity $v'$. There were no external horizontal forces acting on the cart and the rain, so the momentum is still $P$, and the velocity $v'$ is given by $P = (M+m)v'$.

Momentum is taken away from the cart and added to the falling rain by the internal forces acting between the water and the cart.

Think about a different situation where the cart collides with a number of small rocks spaced along the track, and each rock sticks to the cart in a totally inelastic collision. In that case, its should be clear that momentum is transferred from the cart to each rock by the equal and opposite impact forces on the cart and the rocks.

In the original problem, it is hard to visualize exactly what is the force between the cart and the water, but the good thing about conservation of momentum is you don't need to work out all the local details of how the rain water sloshes around inside the cart to find the global behaviour of the system - i.e. the change in speed of the cart.

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The net force on the cart is not $0$. The cart is interacting with the rain in the cart, giving up some of it's momentum to the rain so that it travels the same velocity as the cart.

The net force of the cart and rain system is $0$. Any force the rain puts on the cart is directly opposed by the force of the cart on the rain inside it.

As rain enters the system, it's momentum needs to increase so that it can move with the same horizontal velocity as the cart. This comes from the momentum of the cart, therefore the cart + rain system still conserves momentum and has no net force acting on it (except for the vertical forces of rain falling; but those don't affect horizontal movement here).

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