Two identical balls each of mass m moving on straight track approaching towards each other with same speed. Find kinetic energy of the two ball system is equal to the total energy loss during collision E is the total kinetic energy of the balls before collision and E' is after collision and coefficient of restitution is e. Then choose the correct options.

Options

A. `(E)/(E') = 2`
B. `e=(1)/(2)`
C. `e=(1)/(sqrt(2))`
D. `(E)/(E')=sqrt(2)`

Explanation

To solve the problem step by step, we will analyze the kinetic energy of the two identical balls before and after the collision, as well as the energy lost during the collision.

### Step 1: Determine the initial kinetic energy (E) of the system
Since both balls have the same mass \( m \) and are moving towards each other with the same speed \( u \), the total initial kinetic energy \( E \) of the system can be calculated as follows:

\[ E = \text{K.E. of ball 1} + \text{K.E. of ball 2} = \frac{1}{2} m u^2 + \frac{1}{2} m u^2 = m u^2 \]

### Step 2: Define the coefficient of restitution (e)
The coefficient of restitution \( e \) is defined as the ratio of the relative speed after the collision to the relative speed before the collision. For two identical balls moving towards each other, the relative speed before the collision is \( 2u \) (since both are moving towards each other).

### Step 3: Calculate the final velocities after the collision
Using the coefficient of restitution, the final velocities \( v_1 \) and \( v_2 \) of the two balls after the collision can be expressed as:

\[ v_1 = -eu \quad \text{(ball 1 moves in the opposite direction)} \]
\[ v_2 = eu \quad \text{(ball 2 moves in the opposite direction)} \]

### Step 4: Calculate the final kinetic energy (E') of the system
The total kinetic energy after the collision \( E' \) can be calculated as:

\[ E' = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 = \frac{1}{2} m (-eu)^2 + \frac{1}{2} m (eu)^2 \]
\[ E' = \frac{1}{2} m (e^2 u^2) + \frac{1}{2} m (e^2 u^2) = m e^2 u^2 \]

### Step 5: Calculate the energy lost during the collision (ΔE)
The energy lost during the collision can be calculated as:

\[ \Delta E = E - E' = m u^2 - m e^2 u^2 = m u^2 (1 - e^2) \]

### Step 6: Set the kinetic energy equal to the total energy loss
According to the problem, the kinetic energy of the two-ball system is equal to the total energy loss during the collision:

\[ E = \Delta E \]
\[ m u^2 = m u^2 (1 - e^2) \]

### Step 7: Simplify the equation
Dividing both sides by \( m u^2 \) (assuming \( m \neq 0 \) and \( u \neq 0 \)) gives:

\[ 1 = 1 - e^2 \]

### Step 8: Solve for e
Rearranging the equation gives:

\[ e^2 = 1 - 1 = 0 \]

This indicates that we need to consider the energy loss correctly. The correct relation should be:

\[ m u^2 e^2 = m u^2 (1 - e^2) \]

### Step 9: Rearranging the equation
This leads to:

\[ e^2 + e^2 = 1 \implies 2e^2 = 1 \implies e^2 = \frac{1}{2} \]

### Step 10: Find the value of e
Taking the square root gives:

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

### Conclusion
Thus, the correct options are:
- \( \frac{E}{E'} = 2 \)
- Coefficient of restitution \( e = \frac{1}{\sqrt{2}} \)