A `10 kg` boy standing in a `40 kg` boat floating on water is `20 m` away from the shore of the river. If the boy moves `8 m` on the boat towards the shore, then how far is he from the shore ? (Assume no friction between boat and water).

Options

A. 12.0 m
B. 13.6 m
C. 12.8 m
D. 11.6 m

Explanation

To solve the problem, we need to analyze the situation using the principle of conservation of momentum. Let's break it down step by step:

### Step 1: Understand the Initial Conditions
- The boy has a mass of \( m_1 = 10 \, \text{kg} \).
- The boat has a mass of \( m_2 = 40 \, \text{kg} \).
- Initially, both the boy and the boat are \( 20 \, \text{m} \) away from the shore.

### Step 2: Define the Movement
- The boy moves \( 8 \, \text{m} \) towards the shore on the boat.
- Let \( x \) be the distance the boat moves away from the shore when the boy moves.

### Step 3: Apply Conservation of Momentum
Since there are no external forces acting on the boy-boat system, the momentum of the system is conserved. The initial momentum is zero (as both are at rest), so the final momentum must also be zero.

The equation for conservation of momentum can be expressed as:
\[ m_1 \cdot (8 - x) + m_2 \cdot (-x) = 0 \]
Where:
- \( (8 - x) \) is the distance the boy moves towards the shore minus the distance the boat moves away from the shore.
- The negative sign for the boat indicates that it moves in the opposite direction.

### Step 4: Substitute the Values
Substituting the values of \( m_1 \) and \( m_2 \):
\[ 10 \cdot (8 - x) - 40 \cdot x = 0 \]

### Step 5: Solve for \( x \)
Expanding the equation:
\[ 80 - 10x - 40x = 0 \]
Combining like terms:
\[ 80 - 50x = 0 \]
Solving for \( x \):
\[ 50x = 80 \implies x = \frac{80}{50} = 1.6 \, \text{m} \]

### Step 6: Calculate the Final Distance from the Shore
Initially, the boy was \( 20 \, \text{m} \) from the shore. After moving \( 8 \, \text{m} \) towards the shore and the boat moving \( 1.6 \, \text{m} \) away from the shore, the final distance \( d \) of the boy from the shore is:
\[ d = 20 - 8 + x = 20 - 8 + 1.6 = 13.6 \, \text{m} \]

### Conclusion
Thus, the boy is \( 13.6 \, \text{m} \) away from the shore after he moves \( 8 \, \text{m} \) towards it.

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