A man of `50 kg` is standing at one end on a boat of length `25m` and mass `200kg`.If he starts running and when he reaches the other end, has a velocity `2ms^(-1)` with respect to the boat.The final velocity of the boat is

Options

A. `2/5`
B. `2/3`
C. `8/5`
D. `8/3`

Answer

To solve the problem step by step, we will use the principle of conservation of momentum.

### Step 1: Understand the System
Initially, both the man and the boat are at rest. Therefore, the initial momentum of the system (man + boat) is zero.

### Step 2: Define Variables
- Mass of the man, \( m_m = 50 \, \text{kg} \)
- Mass of the boat, \( m_b = 200 \, \text{kg} \)
- Length of the boat, \( L = 25 \, \text{m} \)
- Velocity of the man with respect to the boat, \( v_{mb} = 2 \, \text{m/s} \)
- Final velocity of the boat with respect to the ground, \( v_b \) (unknown)

### Step 3: Write the Initial Momentum
The initial momentum of the system is:
\[ P_{\text{initial}} = 0 \, \text{(since both are at rest)} \]

### Step 4: Write the Final Momentum
When the man runs to the other end of the boat, his velocity with respect to the ground can be expressed as:
\[ v_m = v_{mb} + v_b \]
where \( v_m \) is the final velocity of the man with respect to the ground.

The final momentum of the system is:
\[ P_{\text{final}} = m_m \cdot v_m + m_b \cdot v_b \]

### Step 5: Apply Conservation of Momentum
According to the conservation of momentum:
\[ P_{\text{initial}} = P_{\text{final}} \]
Thus,
\[ 0 = m_m \cdot (v_{mb} + v_b) + m_b \cdot v_b \]

### Step 6: Substitute Known Values
Substituting the known values into the equation:
\[ 0 = 50 \cdot (2 + v_b) + 200 \cdot v_b \]

### Step 7: Simplify the Equation
Expanding the equation gives:
\[ 0 = 100 + 50v_b + 200v_b \]
\[ 0 = 100 + 250v_b \]

### Step 8: Solve for \( v_b \)
Rearranging the equation to solve for \( v_b \):
\[ 250v_b = -100 \]
\[ v_b = -\frac{100}{250} = -\frac{2}{5} \, \text{m/s} \]

### Conclusion
The final velocity of the boat is:
\[ v_b = -0.4 \, \text{m/s} \]
The negative sign indicates that the boat moves in the opposite direction to the man's running direction.