# Momentum of a falling object

## Main Question or Discussion Point

Hi.

I am studying physics on my own from scratch, so far so good, though I've run into this concept I am struggling to understand.

If we push a large rock over a cliff, it falls because of the pull of the Earth's gravity on it. This force is its weight and it makes the rock accelerate towards the Earth. Its weight does work and the rock gains kinetic energy. It also gains momentum downwards.

Now, according to the book I use - Cambridge International AS and A level Physics - something must be gaining an equal amount of momentum in the opposite (upward) direction. it is the Earth, which starts to move upwards as the rock falls downwards. When the rock hits the ground, its momentum becomes zero. At the same instant, the Earth also stops moving upwards. The rock's momentum cancels out the Earth's momentum. At all times during the rock's fall and crash-landing, momentum has been conserved.

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I understand the principle of conservation of momentum, but I don't quite understand how it relates to this. I am guessing it will have something to do with gravity, but even then, why there has to be an equal amount of momentum in the opposite direction?

Thank you for an explanation.

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A.T.
it is the Earth, which starts to move upwards as the rock falls downwards.
Yes.

Momentum must stay constant in the system (Earth-Rock system). If we start out with a system with zero momentum, it must always have 0 momentum (if there are no external forces). This can be stated mathematically as ##\frac{d\mathbf p}{dt} = 0##.

In order for this to happen, the Earth needs to have ##- \mathbf p## (where the momentum of the rock is ##\mathbf p##) to cancel out the momentum of the rock. If the momentum where different, this would mean that the force is not equal and opposite, and hence we have a change in momentum.

Momentum must stay constant in the system (Earth-Rock system). If we start out with a system with zero momentum, it must always have 0 momentum (if there are no external forces). This can be stated mathematically as ##\frac{d\mathbf p}{dt} = 0##.

In order for this to happen, the Earth needs to have ##- \mathbf p## (where the momentum of the rock is ##\mathbf p##) to cancel out the momentum of the rock. If the momentum where different, this would mean that the force is not equal and opposite, and hence we have a change in momentum.
I understand all of this, but I don't get why the rock should "interact" with the Earth.

The momentum in this book was explained on two colliding objects, so I am struggling to transfer those findings to this example. I mean, if I have a ball rolling towards a wall, the wall clearly doesn't move towards the ball. Yes, I understand that the Earth has gravity which causes the rock to fall towards it, but I can't link these two findings.

I understand all of this, but I don't get why the rock should "interact" with the Earth.

The momentum in this book was explained on two colliding objects, so I am struggling to transfer those findings to this example. I mean, if I have a ball rolling towards a wall, the wall clearly doesn't move towards the ball. Yes, I understand that the Earth has gravity which causes the rock to fall towards it, but I can't link these two findings.
They rock and Earth interact because of gravity. If we have a force ##\mathbf F = \frac{d\mathbf p}{dt}## on the rock, there must be a force ##-\mathbf F= - \frac{d \mathbf p}{dt}## on the Earth. The interaction is through gravity. I'm not sure what you're not understanding.

To use your example of a ball rolling into a wall: Consider a completely elastic collision between a ball and a wall. If the wall rests on a surface without friction, it will have to "recoil" after the collision and take the momentum of the ball, leaving the ball stationary. The interaction is the collision.

tony873004