I've got a question:
Imagine this scenario where someone releases a marble down an inclined ramp and at the end of the ramp a block is fixed. So the marble will roll down the ramp and it will collide with the block.
Since the block is fixed, m(marble)u(marble) + m(block)u(block) = m(marble)v(marble) + m(block)v(block) by conservation of momentum. Since u(block) and v(block) = 0, u(marble) = v(marble).
Yet, KE of the marble is lost and when it rebounds it will not go back up to its original height. How can momentum be conserved in this case?
To clarify, KE of the marble is lost through heat for eg. due to this collision being inelastic. If KE of the marble is lost, then how does u(marble) = v(marble)?
The earth moved under its feet.
In this scenario, although the kinetic energy (KE) of the marble is lost during the collision with the block, momentum is still conserved. Let's break down the explanation step by step:
1. Conservation of momentum: The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act upon it. In this case, the system consists of the marble and the block.
2. Initial momentum: Initially, the marble and the block are at rest, so their velocities (u) are zero. Therefore, the initial momentum of both the marble and the block is zero.
3. Collision: When the marble rolls down the ramp and collides with the block, there is a transfer of momentum from the marble to the block. Due to this transfer, the marble's velocity changes from its initial velocity (u(marble)) to a final velocity (v(marble)). However, since the block is fixed, its velocity (v(block)) remains zero throughout.
4. Conserved momentum: According to the principle of conservation of momentum, the total momentum before the collision must be equal to the total momentum after the collision. Thus, m(marble)u(marble) = m(marble)v(marble) + m(block)v(block) simplifies to m(marble)u(marble) = m(marble)v(marble).
5. Lost kinetic energy: Although momentum is conserved, the marble does lose kinetic energy during the collision. This loss is due to internal forces and other factors like friction and vibrations within the system. The collision between the marble and the block results in a dissipation of energy, which is why the marble does not rebound to its original height.
In conclusion, even though the marble loses kinetic energy, momentum is still conserved. The net change in momentum of the system remains zero, as the momentum gained by the block is equal to the momentum lost by the marble.