A girl of mass 50 kg stands on a railroad car of mass 75 kg moving with velocity `20 ms^(-1)` . Now , the girl begins to run with a velocity of `10ms^(-1)` with respect to the car in the same direction , as that of the car. The velocity of the car at this instant will be

Options

A. `20 ms ^(-1)`
B. `24 ms ^(-1)`
C. `16ms ^(-1)`
D. `18 ms ^(-1)`

Answer

To solve the problem, we will use the principle of conservation of momentum. Here are the steps to find the final velocity of the railroad car after the girl starts running.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- Mass of the girl (m_g) = 50 kg
- Mass of the car (m_c) = 75 kg
- Initial velocity of the car (V_initial) = 20 m/s
- Velocity of the girl with respect to the car (V_g/c) = 10 m/s (in the same direction as the car)

2. **Calculate the Initial Momentum:**
The total initial momentum (P_initial) of the system (girl + car) can be calculated as:
\[ P_{\text{initial}} = (m_g + m_c) \times V_{\text{initial}} = (50 \, \text{kg} + 75 \, \text{kg}) \times 20 \, \text{m/s = 125 \, kg} \times 20 \, \text{m/s} = 2500 \, \text{kg m/s} \]

3. **Determine the Final Velocities:**
Let the final velocity of the car be \( V_f \). The velocity of the girl with respect to the ground (V_g) can be expressed as:
\[ V_g = V_f + V_g/c = V_f + 10 \, \text{m/s} \]

4. **Calculate the Final Momentum:**
The total final momentum (P_final) of the system can be expressed as:
\[ P_{\text{final}} = m_c \times V_f + m_g \times V_g = 75 \, \text{kg} \times V_f + 50 \, \text{kg} \times (V_f + 10 \, \text{m/s}) \]
Simplifying this gives:
\[ P_{\text{final}} = 75 V_f + 50 V_f + 500 = 125 V_f + 500 \]

5. **Set Initial Momentum Equal to Final Momentum:**
According to the conservation of momentum:
\[ P_{\text{initial}} = P_{\text{final}} \]
Therefore, we have:
\[ 2500 = 125 V_f + 500 \]

6. **Solve for \( V_f \):**
Rearranging the equation:
\[ 125 V_f = 2500 - 500 \]
\[ 125 V_f = 2000 \]
\[ V_f = \frac{2000}{125} = 16 \, \text{m/s} \]

### Final Answer:
The final velocity of the car is \( V_f = 16 \, \text{m/s} \).