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 of finding the ratio of velocities of two bodies after a collision, we can follow these steps:

### Step 1: Understand the Problem
We have two bodies of equal mass \( m \). One body is moving with a velocity \( v \) and the other is stationary. After the collision, we need to find the velocities \( v_1 \) and \( v_2 \) of the two bodies, and then determine the ratio \( \frac{v_1}{v_2} \).

### Step 2: Apply Conservation of Momentum
The law of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision.

Before the collision:
- Momentum of the first body = \( mv \)
- Momentum of the second body = \( 0 \)

Total initial momentum \( P_{initial} = mv + 0 = mv \)

After the collision:
- Momentum of the first body = \( mv_1 \)
- Momentum of the second body = \( mv_2 \)

Total final momentum \( P_{final} = mv_1 + mv_2 \)

Setting these equal gives us:
\[ mv = mv_1 + mv_2 \]
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ v = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 3: Apply 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.

Given \( e = \frac{1}{2} \), we can write:
\[ e = \frac{v_2 - v_1}{v - 0} \]
Substituting the value of \( e \):
\[ \frac{1}{2} = \frac{v_2 - v_1}{v} \]
Cross-multiplying gives:
\[ v_2 - v_1 = \frac{v}{2} \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( v = v_1 + v_2 \) (Equation 1)
2. \( v_2 - v_1 = \frac{v}{2} \) (Equation 2)

From Equation 2, we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = v_1 + \frac{v}{2} \]

Substituting \( v_2 \) into Equation 1:
\[ v = v_1 + \left(v_1 + \frac{v}{2}\right) \]
Simplifying this gives:
\[ v = 2v_1 + \frac{v}{2} \]
Rearranging:
\[ 2v_1 = v - \frac{v}{2} = \frac{v}{2} \]
Thus:
\[ v_1 = \frac{v}{4} \]

Now substituting \( v_1 \) back into Equation 2 to find \( v_2 \):
\[ v_2 = v_1 + \frac{v}{2} = \frac{v}{4} + \frac{v}{2} = \frac{v}{4} + \frac{2v}{4} = \frac{3v}{4} \]

### Step 5: Find the Ratio of Velocities
Now we have:
- \( v_1 = \frac{v}{4} \)
- \( v_2 = \frac{3v}{4} \)

The ratio \( \frac{v_1}{v_2} \) is:
\[ \frac{v_1}{v_2} = \frac{\frac{v}{4}}{\frac{3v}{4}} = \frac{1}{3} \]

### Conclusion
The ratio of the velocities of the two bodies after the collision is \( \frac{1}{3} \).