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Velocity wrt COM

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For problem: https://pasteboard.co/IeXpnsn.png, find velocity with which small mass collide with bigger mass, on its return journey.

The solution given as https://pasteboard.co/IeXqMzD.png, I don't understand why wrt Com small sphere will return with same but opposite directed velocity, please explain.

asked May 16, 2019 in Physics Problems by n3 (508 points)
edited May 16, 2019 by sammy gerbil

1 Answer

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Both spheres have their initial velocities reversed when they come into contact again. This is a consequence of both the Conservation of Momentum and the Conservation of Energy.

This is essentially the same as throwing a ball up into the air. If air resistance can be ignored, so that there is no loss of energy, it returns to the Earth with the same speed as it was launched. The Earth also returns to the ball with the same speed at which it initially moved away.

In this case also there is no loss of energy between the initial impulse and the final collision between the spheres. The final gravitational potential energy is the same as the initial potential energy, because the spheres return to their initial positions. So the final total kinetic energy will be the same as the initial total kinetic energy.

In the COM frame the momentum of the big sphere is always equal and opposite to that of the small sphere. (The centre of mass frame is the same as the centre of momentum frame.) Therefore if there are no external forces, so that momentum is conserved, then the speed of the small sphere is always in the same proportion $\frac{M}{m}$ to that of the big sphere, and the KE of the small sphere is also always in the same proportion.

When this consequence of the conservation of momentum is combined with the conservation of energy, it can be deduced that the speeds of both particles will be the same when they return to their initial positions.

If there was a loss of energy due to internal forces the two particles would not have their initial speeds when they returned to the initial position, but because the conservation of momentum still applies the ratio of their two speeds would remain at all times the same as initially, and they would return to the initial position at exactly the same time.

If the ratio of velocities varied, while the total energy was conserved, then one particle would return to the starting point before the other. This would mean that the position of the COM had changed, which cannot happen if there are no external forces.

answered May 16, 2019 by sammy gerbil (28,896 points)
selected May 17, 2019 by n3