A ball of mass 1 kg. moving with velocity 7 m/sec overtakes and collides with a ball of mass 2 kg. moving with velocity 1 m/sec in the same direction. If e = 3 / 4, the velocity of the lighter ball after impact is

Options

A. 6 m/sec
B. 3/2 m/sec
C. 1 m/sec
D. 0 m/sec

Answer

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

### Step 1: Identify the given data
- Mass of the lighter ball (m1) = 1 kg
- Initial velocity of the lighter ball (u1) = 7 m/s
- Mass of the heavier ball (m2) = 2 kg
- Initial velocity of the heavier ball (u2) = 1 m/s
- Coefficient of restitution (e) = 3/4

### Step 2: Apply the conservation of momentum
The total momentum before the collision is equal to the total momentum after the collision.

\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]

Where:
- \(v_1\) = final velocity of the lighter ball
- \(v_2\) = final velocity of the heavier ball

Substituting the known values:

\[ (1 \, \text{kg} \times 7 \, \text{m/s}) + (2 \, \text{kg} \times 1 \, \text{m/s}) = (1 \, \text{kg} \times v_1) + (2 \, \text{kg} \times v_2) \]

This simplifies to:

\[ 7 + 2 = v_1 + 2v_2 \]

\[ 9 = v_1 + 2v_2 \quad \text{(Equation 1)} \]

### 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 the known values:

\[ \frac{3}{4} = \frac{v_2 - v_1}{7 - 1} \]

This simplifies to:

\[ \frac{3}{4} = \frac{v_2 - v_1}{6} \]

Cross-multiplying gives:

\[ 3 \times 6 = 4(v_2 - v_1) \]

\[ 18 = 4v_2 - 4v_1 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations simultaneously
We have two equations:

1. \(9 = v_1 + 2v_2\) (Equation 1)
2. \(18 = 4v_2 - 4v_1\) (Equation 2)

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

\[ v_1 = 9 - 2v_2 \]

Substituting this into Equation 2:

\[ 18 = 4v_2 - 4(9 - 2v_2) \]

Expanding this gives:

\[ 18 = 4v_2 - 36 + 8v_2 \]

Combining like terms:

\[ 18 + 36 = 4v_2 + 8v_2 \]

\[ 54 = 12v_2 \]

Dividing both sides by 12:

\[ v_2 = \frac{54}{12} = 4.5 \, \text{m/s} \]

### Step 5: Find \(v_1\)
Now, substituting \(v_2\) back into the equation for \(v_1\):

\[ v_1 = 9 - 2(4.5) = 9 - 9 = 0 \, \text{m/s} \]

### Final Answer
The velocity of the lighter ball after impact is \(v_1 = 0 \, \text{m/s}\).

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