A ball is dropped from height h on the ground where coefficient of restitution is e. After one balance the maximum height is

We can find the maximum height reached by a ball after it is dropped from height h and bounces on the ground with a coefficient of restitution e using the following analysis:

1. When the ball is dropped from height h, it gains kinetic energy just before hitting the ground. The potential energy at height h is converted to kinetic energy:
Potential Energy (PE) = mgh
Kinetic Energy (KE) at the moment of impact = mgh

2. When it bounces, some energy is lost due to the coefficient of restitution e. The coefficient of restitution is defined as the ratio of the velocity after the bounce to the velocity before the bounce:
e = (velocity after bounce) / (velocity before bounce)

3. Velocity at the instant of hitting the ground (v) is determined by the relation:
v = √(2gh)
4. Just after the bounce, the velocity is given by,
velocity after bounce = e * v = e * √(2gh)
5. Maximum height (h’) attained just after the bounce can be calculated from the K.E. at the instant of bounce
K.E. after bounce = (1/2) m (e * √(2gh))²
This kinetic energy is turned back into potential energy at the maximum height:
PE at max height = mgh’
Thus, (1/2) m (e² * 2gh) = mgh’
Simplifying gives:
h’ = e²h

Final Answer:
The maximum height after the bounce is e²h.

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