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I know that momentum is conserved in inelastic collisions. In other words, if the kinetic energy of the system is converted to potential, the momentum is still conserved

Suppose we have the situation where a ball is rolling up a hill. The ball does not have enough energy to get to the top of the hill, and so, it rolls back down.

The initial momentum was directed towards the hill, and the final momentum is directed away from the hill. How is momentum conserved?

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    $\begingroup$ Effectively a duplicate of Collisions between an object and a wall. The hill is not a fixed object. The hill is attached to the planet Earth, and the whole planet moves (by a tiny amount!) so that momentum is conserved. $\endgroup$ – John Rennie Mar 8 '18 at 5:25
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    $\begingroup$ This is just like when you drop a ball. How does it get all that momentum? Simple the earth moves a tiny amount upward. $\endgroup$ – SmarthBansal Mar 8 '18 at 6:00
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    $\begingroup$ Possible duplicate of Collisions between an object and a wall $\endgroup$ – Chris Mar 9 '18 at 0:08
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Conservation of momentum only applies to isolated systems. In your scenario, the momentum of the ball by itself is changed, and not conserved, because it interacts with the hill. However, you can include the hill in your system if you like. Unfortunately, the hill is attached to the Earth, so you would have to include that too. Alternatively, you can imagine slicing the hill horizontally at ground level and moving the hill to a frozen lake. Assume that the hill is smooth and rigid and can slide without friction on the lake surface. Now the system is comprised of the hill plus the ball, and is isolated from horizontal influences (after the ball starts rolling), and momentum will be conserved. I don't want to describe exactly what will happen in this scenario because it might become apparent to you just by thinking about it.

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