A ball of mass m moving with velocity u collides head-on which the second ball of mass m at rest. If the coefficient of restitution is e and velocity of first ball after collision is `v_(1)` and velocity of second ball after collision is `v_(2)` then

Options

A. `v_(1)=((1-e)u)/(2),v_(2)=((1+e)u)/(2)`
B. `v_(1)=((1+e)u)/(2),v_(2)=((1-e)u)/(2)`
C. `v_(1)=u/2,v_(2)=-u/2`
D. `v_(1)=(1+e)u,v_(2)=(1-e)u`

Answer

To solve the problem of a head-on collision between two balls, we need to apply the principles of conservation of momentum and the definition of the coefficient of restitution. Let's break it down step by step.

### Step 1: Understand the scenario
We have two balls of mass \( m \). The first ball is moving with an initial velocity \( u \), and the second ball is at rest. After the collision, the velocities of the first and second balls are \( v_1 \) and \( v_2 \) respectively.

### Step 2: Apply the conservation of momentum
The total momentum before the collision must equal the total momentum after the collision.

The initial momentum is:
\[ \text{Initial Momentum} = m \cdot u + m \cdot 0 = mu \]

The final momentum is:
\[ \text{Final Momentum} = m \cdot v_1 + m \cdot v_2 \]

Setting these equal gives us:
\[ mu = mv_1 + mv_2 \]

Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ u = v_1 + v_2 \quad \text{(Equation 1)} \]

### 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.

The formula is:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} = \frac{v_2 - v_1}{u - 0} = \frac{v_2 - v_1}{u} \]

Rearranging gives us:
\[ v_2 - v_1 = e \cdot u \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
Now we have two equations:
1. \( u = v_1 + v_2 \) (Equation 1)
2. \( v_2 - v_1 = e \cdot u \) (Equation 2)

From Equation 1, we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = u - v_1 \]

Substituting this into Equation 2:
\[ (u - v_1) - v_1 = e \cdot u \]
\[ u - 2v_1 = e \cdot u \]
\[ u - e \cdot u = 2v_1 \]
\[ (1 - e)u = 2v_1 \]
\[ v_1 = \frac{(1 - e)u}{2} \quad \text{(Equation 3)} \]

### Step 5: Find \( v_2 \)
Now substitute \( v_1 \) back into Equation 1 to find \( v_2 \):
\[ u = \frac{(1 - e)u}{2} + v_2 \]
\[ v_2 = u - \frac{(1 - e)u}{2} \]
\[ v_2 = \frac{2u - (1 - e)u}{2} \]
\[ v_2 = \frac{(1 + e)u}{2} \quad \text{(Equation 4)} \]

### Final Results
The velocities after the collision are:
\[ v_1 = \frac{(1 - e)u}{2} \]
\[ v_2 = \frac{(1 + e)u}{2} \]