A body of 2 kg mass makes an elastic collision with another body at rest. The velocity of the 2 kg mass is reduced to one-fourth of the original velocity. However, direction remains unchanged. The mass of the body struck is `M xx 10^(-1) kg`. Find the value of M.

Explanation

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

### Step 1: Define the Variables
Let:
- Mass of the first body (m1) = 2 kg
- Initial velocity of the first body (u) = u m/s
- Final velocity of the first body after collision (v1) = u/4 m/s
- Mass of the second body (m2) = M × 10^(-1) kg
- Initial velocity of the second body (u2) = 0 m/s (at rest)
- Final velocity of the second body after collision (v2) = v m/s (unknown)

### Step 2: Apply Conservation of Momentum
According to the law of conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
This can be expressed as:
\[ m_1 \cdot u + m_2 \cdot 0 = m_1 \cdot v_1 + m_2 \cdot v_2 \]

Substituting the known values:
\[ 2u + 0 = 2 \cdot \frac{u}{4} + m_2 \cdot v_2 \]
This simplifies to:
\[ 2u = \frac{u}{2} + m_2 \cdot v_2 \]
Multiplying through by 2 to eliminate the fraction:
\[ 4u = u + 2m_2 \cdot v_2 \]
Rearranging gives:
\[ 3u = 2m_2 \cdot v_2 \]
(Equation 1)

### Step 3: Use the Elastic Collision Condition
In an elastic collision, the relative velocity of separation is equal to the relative velocity of approach. This can be expressed as:
\[ v_1 - v_2 = -(u - 0) \]
Substituting the values:
\[ \frac{u}{4} - v_2 = -u \]
Rearranging gives:
\[ v_2 = \frac{u}{4} + u = \frac{5u}{4} \]
(Equation 2)

### Step 4: Substitute Equation 2 into Equation 1
Now, we substitute \( v_2 \) from Equation 2 into Equation 1:
\[ 3u = 2m_2 \cdot \frac{5u}{4} \]
Cancelling \( u \) from both sides (assuming \( u \neq 0 \)):
\[ 3 = 2m_2 \cdot \frac{5}{4} \]
Multiplying both sides by 4:
\[ 12 = 10m_2 \]
Dividing both sides by 10:
\[ m_2 = \frac{12}{10} = 1.2 \text{ kg} \]

### Step 5: Express in the Required Format
Since \( m_2 = M \times 10^{-1} \):
\[ 1.2 = M \times 10^{-1} \]
Thus,
\[ M = 1.2 \times 10 = 12 \]

### Final Answer
The value of \( M \) is **12**.

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