A ball of mass 4 kg moving on a smooth horizontal surface makes an elastic collision with another ball of mass m at rest in the line of motion of first ball. If after collision first ball moves in the same direction with one fourth of its velocity before collision, then mass of second ball is

Options

A. 4 kg
B. 4.4 kg
C. 2.4 kg
D. 2 kg

Answer

To solve the problem step by step, we will use the principles of conservation of momentum and the properties of elastic collisions.

### Step 1: Define the variables
Let's denote:
- Mass of the first ball (m1) = 4 kg
- Mass of the second ball (m2) = m (unknown)
- Initial velocity of the first ball (u1) = u (unknown)
- Final velocity of the first ball (u1') = u/4 (given)
- Initial velocity of the second ball (u2) = 0 (at rest)
- Final velocity of the second ball (u2') = v (unknown)

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

Before the collision:
\[ \text{Total momentum} = m_1 \cdot u_1 + m_2 \cdot u_2 = 4u + 0 = 4u \]

After the collision:
\[ \text{Total momentum} = m_1 \cdot u_1' + m_2 \cdot u_2' = 4 \cdot \frac{u}{4} + m \cdot v = u + mv \]

Setting the two expressions for momentum equal gives:
\[ 4u = u + mv \]

### Step 3: Rearrange the momentum equation
Rearranging the equation:
\[ 4u - u = mv \]
\[ 3u = mv \]
\[ m = \frac{3u}{v} \] (Equation 1)

### Step 4: Use the elastic collision property
In an elastic collision, the coefficient of restitution (e) is 1. This means:
\[ e = \frac{\text{Relative velocity of separation}}{\text{Relative velocity of approach}} \]

For our case:
\[ 1 = \frac{v - \frac{u}{4}}{u} \]

### Step 5: Solve for v
Rearranging the equation:
\[ v - \frac{u}{4} = u \]
\[ v = u + \frac{u}{4} \]
\[ v = \frac{4u}{4} + \frac{u}{4} = \frac{5u}{4} \] (Equation 2)

### Step 6: Substitute v in Equation 1
Now substitute Equation 2 into Equation 1:
\[ m = \frac{3u}{\frac{5u}{4}} \]
\[ m = \frac{3u \cdot 4}{5u} \]
\[ m = \frac{12}{5} \]
\[ m = 2.4 \text{ kg} \]

### Conclusion
The mass of the second ball (m) is 2.4 kg.