A body from height h is dropped. If the coefficient of restitution is e, then calculate the height achieved after one bounce.a)h1 = e2hb)h1 = e4hc)h1 = e hd)h1 = h/eCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

We know that,
u=√2gh.....
Nd coefficient of restitution (e) =v/u
so by putting the value of both the velocities.
we have, e=√2gx/√2gh
so x=e^2h

Answer

When a body falls from a height h, it strikes the ground with a velocity μ = √2gh Let it rebounds with a velocity v and rise to a height h₁

If the successive heights to which the body rebounds again and again are h₂, h₃,...., then

Clearly, h₁ = e² h, h₂ = e⁴ h ... and so on. Similarly, after nth rebound
h_n = (e²ⁿ)h

Answer

Understanding the Problem
When a body is dropped from a height h, it falls under the influence of gravity. Upon hitting the ground, it rebounds to a certain height based on the coefficient of restitution (e), which measures the elasticity of the collision.
Coefficient of Restitution Explained
- The coefficient of restitution (e) is defined as the ratio of the velocity of separation to the velocity of approach.
- For a dropped object, when it hits the ground, it loses some energy due to inelastic collisions, which is quantified by e.
Calculating the Height After the First Bounce
1. Initial Velocity Before Impact:
- When the body is dropped from height h, it attains a velocity v just before impact given by the equation v = sqrt(2gh), where g is the acceleration due to gravity.
2. Velocity After Impact:
- The velocity after the impact will be e times the velocity before impact, i.e., v' = e * v = e * sqrt(2gh).
3. Height Achieved After the First Bounce:
- The height reached after the bounce (h1) can be calculated using the energy conservation principle.
- The kinetic energy just after the bounce converts back into potential energy at the peak height:
- (1/2) * m * (v')^2 = m * g * h1
- Simplifying, we find:
- (1/2) * m * (e * sqrt(2gh))^2 = m * g * h1
- h1 = e^2 * h
Conclusion
Thus, the height achieved after the first bounce is given by:
- Correct Answer: h1 = e^2 * h
This aligns with option 'A' and illustrates how energy loss during the bounce affects the rebound height.

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