A ball collides impinges directly on a similar ball at rest. The first ball is brought to rest after the impact. If half of the kinetic energy is lost by impact, the value of coefficient of restitution `( e)` is

Options

A. `1/2`
B. `1/sqrt(3)`
C. `1/sqrt(2)`
D. `(sqrt(3))/2`

Explanation

To solve the problem step by step, we will analyze the collision of two similar balls and determine the coefficient of restitution (e) based on the given conditions.

### Step 1: Define the initial conditions
Let:
- The mass of each ball = m
- The initial velocity of the first ball (u1) = u
- The initial velocity of the second ball (u2) = 0 (since it is at rest)

### Step 2: Calculate the initial kinetic energy (KE_initial)
The initial kinetic energy of the system is given by:
\[ KE_{\text{initial}} = \frac{1}{2} m u_1^2 + \frac{1}{2} m u_2^2 \]
Since the second ball is at rest (u2 = 0), we have:
\[ KE_{\text{initial}} = \frac{1}{2} m u^2 + 0 = \frac{1}{2} m u^2 \]

### Step 3: Define the final conditions after the collision
After the collision:
- The first ball comes to rest, so its final velocity (v1) = 0
- Let the final velocity of the second ball (v2) = v

### Step 4: Calculate the final kinetic energy (KE_final)
The final kinetic energy of the system is:
\[ KE_{\text{final}} = \frac{1}{2} m v_1^2 + \frac{1}{2} m v_2^2 \]
Substituting the values:
\[ KE_{\text{final}} = 0 + \frac{1}{2} m v^2 = \frac{1}{2} m v^2 \]

### Step 5: Determine the change in kinetic energy
The change in kinetic energy (ΔKE) is given by:
\[ \Delta KE = KE_{\text{final}} - KE_{\text{initial}} = \frac{1}{2} m v^2 - \frac{1}{2} m u^2 \]

### Step 6: Relate the kinetic energy loss to the problem statement
According to the problem, half of the kinetic energy is lost:
\[ \Delta KE = \frac{1}{2} KE_{\text{initial}} = \frac{1}{2} \left(\frac{1}{2} m u^2\right) = \frac{1}{4} m u^2 \]
Thus, we can set up the equation:
\[ \frac{1}{2} m v^2 - \frac{1}{2} m u^2 = -\frac{1}{4} m u^2 \]

### Step 7: Simplify the equation
Dividing through by \(\frac{1}{2} m\):
\[ v^2 - u^2 = -\frac{1}{2} u^2 \]
Rearranging gives:
\[ v^2 = \frac{1}{2} u^2 \]

### Step 8: Solve for v
Taking the square root:
\[ v = \frac{u}{\sqrt{2}} \]

### Step 9: Calculate the coefficient of restitution (e)
The coefficient of restitution is defined as:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting the known values:
\[ e = \frac{v - 0}{u - 0} = \frac{v}{u} \]
Substituting \(v = \frac{u}{\sqrt{2}}\):
\[ e = \frac{\frac{u}{\sqrt{2}}}{u} = \frac{1}{\sqrt{2}} \]

### Final Answer
The value of the coefficient of restitution \(e\) is:
\[ \boxed{\frac{1}{\sqrt{2}}} \]