A body of mass `2 kg` makes an elastic head - on collision another body at rest and continues to move in the original direction with one fourth of its original speed . The mass of the second body which collides with the first body is

Options

A. `2 kg`
B. `1.2 kg`
C. `3 kg`
D. `1.5 kg`

Explanation

To solve the problem, we will use the principles of conservation of momentum and the properties of elastic collisions. Let's break it down step by step.

### Step 1: Understand the Initial Conditions
We have:
- Mass of the first body (m1) = 2 kg
- Initial velocity of the first body (u1) = V0 (unknown)
- Mass of the second body (m2) = M (unknown)
- Initial velocity of the second body (u2) = 0 (at rest)

### Step 2: Determine Final Velocities
After the collision:
- The first body continues to move in the original direction with one fourth of its original speed.
- Final velocity of the first body (v1) = V0 / 4
- Let the final velocity of the second body (v2) = V (unknown)

### Step 3: Apply Conservation of Momentum
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]

Initial momentum:
\[ p_{\text{initial}} = m_1 \cdot u_1 + m_2 \cdot u_2 = 2 \cdot V0 + M \cdot 0 = 2V0 \]

Final momentum:
\[ p_{\text{final}} = m_1 \cdot v_1 + m_2 \cdot v_2 = 2 \cdot \frac{V0}{4} + M \cdot V = \frac{V0}{2} + M \cdot V \]

Setting initial momentum equal to final momentum:
\[ 2V0 = \frac{V0}{2} + M \cdot V \]

### Step 4: Rearranging the Equation
Rearranging the equation gives:
\[ M \cdot V = 2V0 - \frac{V0}{2} \]
\[ M \cdot V = \frac{4V0}{2} - \frac{V0}{2} = \frac{3V0}{2} \]

### Step 5: Use Elastic Collision Condition
For elastic collisions, the following equation holds:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Where \( e = 1 \) for perfectly elastic collisions.

Substituting the values:
\[ 1 = \frac{V - \frac{V0}{4}}{V0 - 0} \]

### Step 6: Solve for V
This simplifies to:
\[ V - \frac{V0}{4} = V0 \]
\[ V = V0 + \frac{V0}{4} = \frac{5V0}{4} \]

### Step 7: Substitute V Back into the Momentum Equation
Now substitute \( V = \frac{5V0}{4} \) back into the momentum equation:
\[ M \cdot \frac{5V0}{4} = \frac{3V0}{2} \]

### Step 8: Solve for M
Now, we can solve for M:
\[ M = \frac{\frac{3V0}{2}}{\frac{5V0}{4}} \]
\[ M = \frac{3V0}{2} \cdot \frac{4}{5V0} \]
\[ M = \frac{12}{10} = \frac{6}{5} \text{ kg} = 1.2 \text{ kg} \]

### Final Answer
The mass of the second body is **1.2 kg**.

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