A sphere impinges directly on an equal sphere at rest. If the coefficient of restitution is 1/2, then their velocities after the impact are in the ratio

Options

A. `1:2`
B. `2:3`
C. `1:3`
D. `3:4`

Answer

To solve the problem of two equal spheres colliding, we will use the principles of conservation of momentum and the coefficient of restitution.

### Step 1: Define the Variables
Let:
- \( m \) = mass of each sphere (since they are equal)
- \( u_1 \) = initial velocity of the first sphere (before impact)
- \( u_2 \) = initial velocity of the second sphere (at rest, so \( u_2 = 0 \))
- \( v_1 \) = final velocity of the first sphere (after impact)
- \( v_2 \) = final velocity of the second sphere (after impact)
- \( e \) = coefficient of restitution = \( \frac{1}{2} \)

### Step 2: Apply Conservation of Momentum
According to the law of conservation of momentum:
\[ m u_1 + m u_2 = m v_1 + m v_2 \]
Since \( u_2 = 0 \), the equation simplifies to:
\[ m u_1 = m v_1 + m v_2 \]
Dividing through by \( m \):
\[ u_1 = v_1 + v_2 \quad (1) \]

### Step 3: Apply the Coefficient of Restitution
The coefficient of restitution is defined as:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting \( u_2 = 0 \):
\[ e = \frac{v_2 - v_1}{u_1} \]
Substituting \( e = \frac{1}{2} \):
\[ \frac{1}{2} = \frac{v_2 - v_1}{u_1} \]
Rearranging gives us:
\[ v_2 - v_1 = \frac{1}{2} u_1 \quad (2) \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( u_1 = v_1 + v_2 \)
2. \( v_2 - v_1 = \frac{1}{2} u_1 \)

From equation (2), we can express \( v_2 \) in terms of \( v_1 \):
\[ v_2 = v_1 + \frac{1}{2} u_1 \quad (3) \]

Substituting equation (3) into equation (1):
\[ u_1 = v_1 + \left(v_1 + \frac{1}{2} u_1\right) \]
This simplifies to:
\[ u_1 = 2v_1 + \frac{1}{2} u_1 \]
Rearranging gives:
\[ u_1 - \frac{1}{2} u_1 = 2v_1 \]
\[ \frac{1}{2} u_1 = 2v_1 \]
Thus:
\[ u_1 = 4v_1 \quad (4) \]

### Step 5: Find the Ratio of Velocities
Now substituting \( u_1 = 4v_1 \) back into equation (1):
\[ 4v_1 = v_1 + v_2 \]
This gives:
\[ v_2 = 4v_1 - v_1 = 3v_1 \]

### Step 6: Write the Final Ratio
The final velocities after the impact are:
- \( v_1 \) (velocity of the first sphere)
- \( v_2 = 3v_1 \) (velocity of the second sphere)

Thus, the ratio of their velocities after the impact is:
\[ \frac{v_1}{v_2} = \frac{v_1}{3v_1} = \frac{1}{3} \]

### Final Answer
The ratio of their velocities after the impact is \( 1:3 \).

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