A man weighing 80 kg is standing in a trolley weighing 320 kg. The trolley is resting on frictionless horizontal rails. If the man starts walking on the trolley with a speed of 1 m/s, then after 4 sec his displacement relative to the ground will be

Options

A. `5` m
B. `4.8` m
C. `3.2` m
D. `3.0` m

Explanation

To solve the problem step by step, we will analyze the situation and apply the principles of conservation of momentum and relative motion.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Weight of the man, \( m_1 = 80 \, \text{kg} \)
- Weight of the trolley, \( m_2 = 320 \, \text{kg} \)
- Speed of the man relative to the trolley, \( v_{mt} = 1 \, \text{m/s} \)
- Time, \( t = 4 \, \text{s} \)

2. **Calculate the Distance Traveled by the Man Relative to the Trolley:**
- The distance \( d_m \) traveled by the man relative to the trolley in 4 seconds is given by:
\[ d_m = v_{mt} \times t = 1 \, \text{m/s} \times 4 \, \text{s} = 4 \, \text{m} \]

3. **Determine the Movement of the Trolley:**
- Since the trolley is on frictionless rails, when the man walks, the trolley will move in the opposite direction to conserve momentum.
- Let \( v_t \) be the speed of the trolley after the man starts walking. By conservation of momentum:
\[ m_1 \cdot v_{mt} + m_2 \cdot (-v_t) = 0 \]
\[ 80 \cdot 1 - 320 \cdot v_t = 0 \]
\[ 80 = 320 \cdot v_t \]
\[ v_t = \frac{80}{320} = 0.25 \, \text{m/s} \]

4. **Calculate the Distance Traveled by the Trolley in 4 Seconds:**
- The distance \( d_t \) traveled by the trolley in 4 seconds is:
\[ d_t = v_t \times t = 0.25 \, \text{m/s} \times 4 \, \text{s} = 1 \, \text{m} \]

5. **Determine the Net Displacement of the Man Relative to the Ground:**
- The man walks 4 m relative to the trolley, but the trolley moves 1 m in the opposite direction. Therefore, the total displacement \( d_{net} \) of the man relative to the ground is:
\[ d_{net} = d_m - d_t = 4 \, \text{m} - 1 \, \text{m} = 3 \, \text{m} \]

### Final Answer:
The displacement of the man relative to the ground after 4 seconds is **3 m**.

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