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In this stack and this other stack which I had asked, there was discussion on when you can apply momentum conservation in collisions. The conclusion was that you can do it if time period of collision is small.

However recently I came across a question in which momentum conservation is not obeyed even if time period of collision is small.

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So, in the set up of image above, a block of mass M is dropped from top of a ramp where it collides perfectly inelastically with a second ramp at 'B'. Supposedly, the momentum is conserved in direction along the surface and not in the direction perpendicular to it.

I was confused by this because, we used the collisions have small time period argument to argue about momentum in the first case but not here. So, why exactly can't I use the argument that collision time period is small to say that momentum should be conserved in all directions here?

tryst with freedom
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"The conclusion was that you can do it if time period of collision is small". Here you blatantly ignore other assumptions regarding momentum conservation of colliding bodies which though not mentioned in your previous discussion, still need to hold true for momentum conservation. That assumption is that the concerned force should be "non-impulsive" and if it is "impulsive" it should be internal to our chosen system of particles. Here "impulsive" is a relative term and refers to sudden forces which change momentum appreciably even if collision times are negligible because thier larger magnitude of force makes the product F.dt i.e. impulse, considerable. Collision forces are in general impulsive, like the normal force a body exerts on other. Imagine dropping a ball to ground. Though its collision with ground is very fleeting, its momentum changes, infact reverses. This is beacuse the normal acting on ball from ground is an impulsive force. When considering two colliding bodies the normal reaction is still impulsive but as it acts equally and oppositely on both it becomes an internal force. The whole point of making time period small was to prevent the " non impulsive forces " from contributing to momentum change. So as long as there are no external impulsive forces on our chosen system the quoted statement holds true.

Now for the given scenario the normal force from incline is impulsive disallowing momentum conservation.

Prakhar Pratap Mall
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