A ball is dropped on a horizontal surface from height h. If it rebounds upto `h/2` after first collision then coefficient of restitution between ball and surface is

Options

A. `1 sqrt2`
B. `1/2`
C. `1/4`
D. `1/2 + sqrt2`

Explanation

To find the coefficient of restitution (e) between a ball and a horizontal surface when the ball is dropped from a height \( h \) and rebounds to a height of \( \frac{h}{2} \), we can follow these steps:

### Step 1: Determine the velocity of the ball just before impact
When the ball is dropped from height \( h \), we can use the third equation of motion to find the velocity just before it hits the ground. The equation is:

\[ v^2 = u^2 + 2gh \]

Here, \( u = 0 \) (initial velocity), \( g \) is the acceleration due to gravity, and \( h \) is the height from which the ball is dropped.

Substituting the values, we get:

\[ v^2 = 0 + 2gh \implies v = \sqrt{2gh} \]

This velocity \( v \) is the velocity of approach (\( V_A \)) just before the ball hits the ground.

### Step 2: Determine the velocity of the ball just after rebound
After rebounding to a height of \( \frac{h}{2} \), we can again use the third equation of motion to find the velocity just after the ball leaves the ground. The equation is:

\[ v^2 = u^2 - 2gh \]

In this case, the final velocity \( v = 0 \) (at the peak of the rebound), and we need to find the initial velocity \( u \) just after the rebound. The height is \( \frac{h}{2} \).

Substituting the values, we have:

\[ 0 = u^2 - 2g\left(\frac{h}{2}\right) \implies u^2 = gh \implies u = \sqrt{gh} \]

This velocity \( u \) is the velocity of separation (\( V_S \)) just after the ball rebounds.

### Step 3: Calculate the coefficient of restitution
The coefficient of restitution \( e \) is defined as the ratio of the velocity of separation to the velocity of approach:

\[ e = \frac{V_S}{V_A} \]

Substituting the values we found:

\[ e = \frac{\sqrt{gh}}{\sqrt{2gh}} = \frac{1}{\sqrt{2}} \]

### Final Answer
The coefficient of restitution between the ball and the surface is:

\[ e = \frac{1}{\sqrt{2}} \]

---