Two elastic spheres impinge directly with equal and opposite velocities. The ratio of their masses so that one of them may be reduced to rest by impact, the coefficient of restitution being is …

Answer

To solve the problem of two elastic spheres colliding directly with equal and opposite velocities, we need to analyze the situation using the principles of conservation of momentum and the coefficient of restitution.

### Step-by-Step Solution:

1. **Define Variables**:
Let:
- \( m_1 \) = mass of sphere 1
- \( m_2 \) = mass of sphere 2
- \( u_1 \) = initial velocity of sphere 1 (moving towards sphere 2)
- \( u_2 \) = initial velocity of sphere 2 (moving towards sphere 1)
- \( v_1 \) = final velocity of sphere 1
- \( v_2 \) = final velocity of sphere 2
- \( e \) = coefficient of restitution

Given that the spheres have equal and opposite velocities, we can set:
\[ u_1 = v \quad \text{and} \quad u_2 = -v \]

2. **Apply Conservation of Momentum**:
According to the law of conservation of momentum:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
Substituting the values of \( u_1 \) and \( u_2 \):
\[ m_1 v + m_2 (-v) = m_1 v_1 + m_2 v_2 \]
Simplifying this gives:
\[ (m_1 - m_2)v = m_1 v_1 + m_2 v_2 \quad \text{(1)} \]

3. **Apply Coefficient of Restitution**:
The coefficient of restitution \( e \) is defined as:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting the values:
\[ e = \frac{v_2 - v_1}{v - (-v)} = \frac{v_2 - v_1}{2v} \]
Rearranging gives:
\[ v_2 - v_1 = 2ev \quad \text{(2)} \]

4. **Substituting Equation (2) into Equation (1)**:
From Equation (2), we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = v_1 + 2ev \]
Substituting this into Equation (1):
\[ (m_1 - m_2)v = m_1 v_1 + m_2 (v_1 + 2ev) \]
Expanding this gives:
\[ (m_1 - m_2)v = (m_1 + m_2)v_1 + 2m_2ev \]
Rearranging terms leads to:
\[ (m_1 - m_2)v - 2m_2ev = (m_1 + m_2)v_1 \]

5. **Setting \( v_1 = 0 \) for Sphere 1 to be at Rest**:
To find the ratio of the masses such that sphere 1 comes to rest after the collision, we set \( v_1 = 0 \):
\[ (m_1 - m_2)v - 2m_2ev = 0 \]
Rearranging gives:
\[ (m_1 - m_2)v = 2m_2ev \]
Dividing through by \( v \) (assuming \( v \neq 0 \)):
\[ m_1 - m_2 = 2m_2e \]
Thus:
\[ m_1 = m_2(1 + 2e) \]

6. **Finding the Ratio of Masses**:
The ratio of the masses \( \frac{m_1}{m_2} \) is:
\[ \frac{m_1}{m_2} = 1 + 2e \]

### Final Answer:
The ratio of their masses so that one of them may be reduced to rest by impact, given the coefficient of restitution \( e \), is:
\[ \frac{m_1}{m_2} = 1 + 2e \]