A body of mass m moving with a constant velocity collides head on with another stationary body of same mass if the coefficient of restitution between the bodies is `(1)/(2)` then ratio of velocities of two bodies after collision with be

Options

A. `(1)/(3)`
B. `(1)/(2)`
C. `(1)/(4)`
D. `1`

Answer

To solve the problem, we need to find the ratio of the velocities of two bodies after a head-on collision, given that the coefficient of restitution is \( \frac{1}{2} \).

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Let the mass of both bodies be \( m \).
- The first body is moving with an initial velocity \( u \).
- The second body is stationary, so its initial velocity is \( 0 \).

2. **Define the Coefficient of Restitution:**
- The coefficient of restitution \( e \) is defined as the ratio of the relative velocity of separation to the relative velocity of approach.
- Mathematically, it is given by:
\[ e = \frac{V_2 - V_1}{u - 0} \]
- Here, \( V_1 \) is the final velocity of the first body, and \( V_2 \) is the final velocity of the second body after the collision.

3. **Substituting the Coefficient of Restitution:**
- Given that \( e = \frac{1}{2} \), we can write:
\[ \frac{1}{2} = \frac{V_2 - V_1}{u} \]
- Rearranging gives:
\[ V_2 - V_1 = \frac{u}{2} \quad \text{(Equation 1)} \]

4. **Conservation of Momentum:**
- Since there are no external forces acting on the system, the momentum before and after the collision must be conserved.
- The initial momentum is:
\[ mu + 0 = mu \]
- The final momentum is:
\[ mV_1 + mV_2 \]
- Setting the initial momentum equal to the final momentum gives:
\[ mu = mV_1 + mV_2 \]
- Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ u = V_1 + V_2 \quad \text{(Equation 2)} \]

5. **Solving the Equations:**
- We now have two equations:
1. \( V_2 - V_1 = \frac{u}{2} \)
2. \( u = V_1 + V_2 \)

- From Equation 1, we can express \( V_2 \) in terms of \( V_1 \):
\[ V_2 = V_1 + \frac{u}{2} \]

- Substitute \( V_2 \) into Equation 2:
\[ u = V_1 + \left(V_1 + \frac{u}{2}\right) \]
\[ u = 2V_1 + \frac{u}{2} \]
\[ u - \frac{u}{2} = 2V_1 \]
\[ \frac{u}{2} = 2V_1 \]
\[ V_1 = \frac{u}{4} \]

- Now substitute \( V_1 \) back into the expression for \( V_2 \):
\[ V_2 = \frac{u}{4} + \frac{u}{2} = \frac{u}{4} + \frac{2u}{4} = \frac{3u}{4} \]

6. **Finding the Ratio of Velocities:**
- Now we can find the ratio of the velocities after the collision:
\[ \frac{V_1}{V_2} = \frac{\frac{u}{4}}{\frac{3u}{4}} = \frac{1}{3} \]

### Final Answer:
The ratio of the velocities of the two bodies after the collision is:
\[ \frac{V_1}{V_2} = \frac{1}{3} \]