A sphere strikes a wall and rebounds with coefficient of restitution `(1)/(3)`. If it rebounds with a velocity of `0.1 m//sec` at an angle of `60^(@)` to the normal to the wall, the loss of kinetic energy is :

Options

A. `50%`
B. `33(1)/(3)%`
C. `40%`
D. `66(2)/(3)%`

Answer

To solve the problem of finding the loss of kinetic energy when a sphere strikes a wall and rebounds, we can follow these steps:

### Step 1: Understand the Given Data
- Coefficient of restitution \( e = \frac{1}{3} \)
- Final velocity \( V = 0.1 \, \text{m/s} \)
- Angle of rebound \( \theta = 60^\circ \)

### Step 2: Break Down the Final Velocity into Components
The final velocity can be broken down into its horizontal and vertical components:
- Horizontal component: \( V_x = V \cos(60^\circ) = 0.1 \cos(60^\circ) = 0.1 \times \frac{1}{2} = 0.05 \, \text{m/s} \)
- Vertical component: \( V_y = V \sin(60^\circ) = 0.1 \sin(60^\circ) = 0.1 \times \frac{\sqrt{3}}{2} = 0.05\sqrt{3} \, \text{m/s} \)

### Step 3: Use the Coefficient of Restitution
The coefficient of restitution relates the velocities before and after the collision:
\[ e = \frac{V_x}{U_x} \]
Where \( U_x \) is the initial horizontal component of the velocity before the collision. Rearranging gives:
\[ U_x = \frac{V_x}{e} = \frac{0.05}{\frac{1}{3}} = 0.15 \, \text{m/s} \]

### Step 4: Determine the Initial Vertical Component
Since the vertical component does not change during the collision:
\[ U_y = V_y = 0.05\sqrt{3} \, \text{m/s} \]

### Step 5: Calculate the Initial Velocity
Now, we can find the magnitude of the initial velocity \( U \):
\[ U = \sqrt{U_x^2 + U_y^2} = \sqrt{(0.15)^2 + (0.05\sqrt{3})^2} \]
Calculating this:
\[ U = \sqrt{0.0225 + 0.0075} = \sqrt{0.03} = 0.1732 \, \text{m/s} \]

### Step 6: Calculate the Initial and Final Kinetic Energies
The kinetic energy is given by:
\[ KE = \frac{1}{2} m v^2 \]
- Initial kinetic energy \( KE_i = \frac{1}{2} m U^2 = \frac{1}{2} m (0.1732)^2 \)
- Final kinetic energy \( KE_f = \frac{1}{2} m V^2 = \frac{1}{2} m (0.1)^2 \)

### Step 7: Find the Loss of Kinetic Energy
The loss of kinetic energy \( \Delta KE \) is given by:
\[ \Delta KE = KE_i - KE_f = \frac{1}{2} m (0.1732^2 - 0.1^2) \]
Calculating the values:
\[ \Delta KE = \frac{1}{2} m (0.03 - 0.01) = \frac{1}{2} m (0.02) = 0.01 m \]

### Conclusion
The loss of kinetic energy is \( 0.01 m \) joules, where \( m \) is the mass of the sphere.