A sphere impinges directly on an equal sphere at rest. If the coefficient of restitution is e, their velocities after the impact are as

Options

A. `1:e`
B. `e:1`
C. `1+e:1-e`
D. `1-e:1+e`

Answer

To solve the problem of two equal spheres colliding, we can follow these steps:

### Step 1: Understand the situation
We have two equal spheres, one moving with an initial velocity \( v \) and the other at rest. We need to find their velocities after the impact, given the coefficient of restitution \( e \).

### Step 2: Define the variables
Let:
- Mass of each sphere = \( m \)
- Initial velocity of the moving sphere = \( v_1 = v \)
- Initial velocity of the stationary sphere = \( v_2 = 0 \)
- Final velocity of the moving sphere after impact = \( v_1' \)
- Final velocity of the stationary sphere after impact = \( v_2' \)

### Step 3: Apply the law of conservation of momentum
According to the law of conservation of momentum, the total momentum before the collision equals the total momentum after the collision:

\[ m v + m \cdot 0 = m v_1' + m v_2' \]

This simplifies to:

\[ v = v_1' + v_2' \quad \text{(Equation 1)} \]

### Step 4: 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:

\[ e = \frac{v_2' - v_1'}{v - 0} \]

This can be rearranged to:

\[ v_2' - v_1' = e v \quad \text{(Equation 2)} \]

### Step 5: Solve the equations
We now have two equations:
1. \( v = v_1' + v_2' \)
2. \( v_2' - v_1' = e v \)

From Equation 1, we can express \( v_2' \) in terms of \( v_1' \):

\[ v_2' = v - v_1' \]

Substituting this into Equation 2:

\[ (v - v_1') - v_1' = e v \]

This simplifies to:

\[ v - 2v_1' = e v \]

Rearranging gives:

\[ 2v_1' = v(1 - e) \]

Thus, we find:

\[ v_1' = \frac{v(1 - e)}{2} \]

Now, substituting \( v_1' \) back into Equation 1 to find \( v_2' \):

\[ v_2' = v - v_1' = v - \frac{v(1 - e)}{2} \]

This simplifies to:

\[ v_2' = \frac{v(1 + e)}{2} \]

### Final Result
The final velocities after the impact are:
- For the moving sphere: \( v_1' = \frac{v(1 - e)}{2} \)
- For the stationary sphere: \( v_2' = \frac{v(1 + e)}{2} \)