A sphere of mass m moving with velocity v collides head-on with another sphere of the same mass at rest. If the coefficient of resistitution `e = 1//2`, then what is the ratio of final velocity of the second sphere to the intial velocity of the first sphere ?

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand 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 can be expressed as:
\[ e = \frac{\text{Velocity of separation}}{\text{Velocity of approach}} \]

### Step 2: Identify Initial Conditions
In this scenario:
- The first sphere has mass \( m \) and an initial velocity \( v \).
- The second sphere has mass \( m \) and is initially at rest (velocity = 0).

### Step 3: Determine the Velocity of Approach
Since the second sphere is at rest, the velocity of approach is simply:
\[ \text{Velocity of approach} = v - 0 = v \]

### Step 4: Write the Equation for Coefficient of Restitution
Using the definition of the coefficient of restitution, we can write:
\[ e = \frac{v_2 - v_1}{v} \]
where:
- \( v_1 \) is the final velocity of the first sphere after the collision.
- \( v_2 \) is the final velocity of the second sphere after the collision.

### Step 5: Apply Conservation of Linear Momentum
According to the conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
This gives us:
\[ mv + 0 = mv_1 + mv_2 \]
Simplifying this, we get:
\[ v = v_1 + v_2 \quad \text{(Equation 1)} \]

### Step 6: Substitute the Coefficient of Restitution
From the coefficient of restitution equation, we can express \( v_1 \):
\[ v_2 - v_1 = e \cdot v \]
Substituting \( e = \frac{1}{2} \):
\[ v_2 - v_1 = \frac{1}{2} v \quad \text{(Equation 2)} \]

### Step 7: Solve the Equations Simultaneously
Now we have two equations:
1. \( v = v_1 + v_2 \)
2. \( v_2 - v_1 = \frac{1}{2} v \)

From Equation 2, we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = v_1 + \frac{1}{2} v \]

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

### Step 8: Find \( v_2 \)
Now substituting \( v_1 \) back into the expression for \( v_2 \):
\[ v_2 = v_1 + \frac{1}{2} v = \frac{1}{4} v + \frac{1}{2} v = \frac{1}{4} v + \frac{2}{4} v = \frac{3}{4} v \]

### Step 9: Calculate the Ratio
The ratio of the final velocity of the second sphere to the initial velocity of the first sphere is:
\[ \frac{v_2}{v} = \frac{\frac{3}{4} v}{v} = \frac{3}{4} \]

### Final Answer
Thus, the ratio of the final velocity of the second sphere to the initial velocity of the first sphere is:
\[ \frac{3}{4} \]