A boat of length `10m` and mass `450kg` is floating without motion in still water. A man of mass `50kg` standing at one end of it walks to the other end of it and stops. The magnitude of the displacement of the boat in metres relative to ground is

Options

A. (a) zero
B. (b) `1m`
C. (c) `2m`
D. (d) `5m`

Explanation

To solve the problem, we will use the principle of conservation of the center of mass. The center of mass of the system (the boat and the man) must remain in the same position since there are no external forces acting on the system.

### Step-by-Step Solution:

1. **Identify the System**:
- We have a boat of mass \( m_1 = 450 \, \text{kg} \) and a man of mass \( m_2 = 50 \, \text{kg} \).
- The length of the boat is \( L = 10 \, \text{m} \).

2. **Initial Position of the Center of Mass**:
- Initially, the man is at one end of the boat. We can set the position of the boat's left end (where the man starts) as \( x = 0 \) and the right end as \( x = 10 \, \text{m} \).
- The initial position of the man is at \( x_1 = 0 \) and the center of mass of the boat is at \( x_{boat} = 5 \, \text{m} \) (the midpoint).

3. **Calculate Initial Center of Mass**:
\[ x_{cm, initial} = \frac{m_1 \cdot x_{boat} + m_2 \cdot x_1}{m_1 + m_2} = \frac{450 \cdot 5 + 50 \cdot 0}{450 + 50} = \frac{2250}{500} = 4.5 \, \text{m} \]

4. **Final Position of the Man**:
- When the man walks to the other end of the boat, his final position is \( x_2 = 10 \, \text{m} \) relative to the boat.
- If the boat moves a distance \( x \) to the left, the man's position relative to the ground will be \( (10 - x) \).

5. **Calculate Final Center of Mass**:
- The new position of the center of mass must remain the same as the initial position:
\[ x_{cm, final} = \frac{m_1 \cdot (5 - x) + m_2 \cdot (10 - x)}{m_1 + m_2} \]
- Setting the initial and final center of mass equal:
\[ 4.5 = \frac{450 \cdot (5 - x) + 50 \cdot (10 - x)}{500} \]

6. **Solve for \( x \)**:
\[ 4.5 \cdot 500 = 450 \cdot (5 - x) + 50 \cdot (10 - x) \]
\[ 2250 = 2250 - 450x + 500 - 50x \]
\[ 2250 = 2250 - 500x \]
\[ 500x = 500 \]
\[ x = 1 \, \text{m} \]

### Conclusion:
The magnitude of the displacement of the boat relative to the ground is \( 1 \, \text{m} \).

---