A sphere of mass m moving with a constant velocity u hits another stationary sphere of the same mass. If e is the coefficient of restitution, then ratio of velocities of the two spheres after collision will be

Options

A. `(1-e)/(1 + e)`
B. `(1 + e)/(1 -e)`
C. `(e + 1)/(e-1)`
D. `(e-1)/(e + 1)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Conditions
We have two spheres of mass \( m \):
- Sphere 1 (moving) has an initial velocity \( u_1 = u \).
- Sphere 2 (stationary) has an initial velocity \( u_2 = 0 \).

### Step 2: Apply Conservation of Momentum
The total momentum before the collision must equal the total momentum after the collision. The initial momentum \( p_i \) is given by:
\[ p_i = m \cdot u + m \cdot 0 = mu \]
After the collision, let the velocities of the two spheres be \( v_1 \) (for sphere 1) and \( v_2 \) (for sphere 2). The final momentum \( p_f \) is:
\[ p_f = m \cdot v_1 + m \cdot v_2 \]
Setting the initial momentum equal to the final momentum gives us:
\[ mu = mv_1 + mv_2 \]
Dividing through by \( m \) (since \( m \neq 0 \)):
\[ u = v_1 + v_2 \quad \text{(Equation 1)} \]

### 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. This can be expressed as:
\[ e = \frac{v_2 - v_1}{u - 0} = \frac{v_2 - v_1}{u} \]
Rearranging gives us:
\[ v_2 - v_1 = eu \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations Simultaneously
Now we have two equations:
1. \( u = v_1 + v_2 \)
2. \( v_2 - v_1 = eu \)

From Equation 1, we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = u - v_1 \]
Substituting this into Equation 2:
\[ (u - v_1) - v_1 = eu \]
This simplifies to:
\[ u - 2v_1 = eu \]
Rearranging gives:
\[ 2v_1 = u - eu \]
\[ v_1 = \frac{u(1 - e)}{2} \quad \text{(Equation 3)} \]

Now substituting \( v_1 \) back into the expression for \( v_2 \):
\[ v_2 = u - v_1 = u - \frac{u(1 - e)}{2} = \frac{u(2 - (1 - e))}{2} = \frac{u(1 + e)}{2} \quad \text{(Equation 4)} \]

### Step 5: Find the Ratio of Velocities
Now we can find the ratio of the velocities \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\frac{u(1 - e)}{2}}{\frac{u(1 + e)}{2}} = \frac{1 - e}{1 + e} \]

### Final Result
Thus, the ratio of the velocities of the two spheres after the collision is:
\[ \frac{v_1}{v_2} = \frac{1 - e}{1 + e} \]