A man weighing 60 kg is standing on a trolley weighing 240 kg. The trolley is resting on a frictionless horizontal rails. If the man starts walking on the trolley along the rails at speed 1m/s, then after 4 seconds, his displacement relative to the ground will be

Options

A. 6 m
B. 4.8 m
C. 3.2 m
D. 2.4 m

Explanation

To solve the problem step by step, we will use the principle of conservation of momentum. Here’s how we can approach it:

### Step 1: Identify the masses and velocities
- Mass of the man (m1) = 60 kg
- Mass of the trolley (m2) = 240 kg
- Velocity of the man relative to the trolley (v1) = 1 m/s (forward)
- Velocity of the trolley (v2) = ? (backward, to be determined)

### Step 2: Apply the conservation of momentum
Since the trolley is on frictionless rails, the total momentum before the man starts walking must equal the total momentum after he starts walking. Initially, both the man and the trolley are at rest, so the initial momentum is 0.

When the man starts walking, the momentum can be expressed as:
\[ m_1 \cdot v_1 + m_2 \cdot v_2 = 0 \]

Substituting the known values:
\[ 60 \cdot 1 + 240 \cdot (-v) = 0 \]
Here, we take the velocity of the trolley as negative because it moves in the opposite direction to the man.

### Step 3: Solve for the trolley's velocity (v2)
Rearranging the equation:
\[ 60 - 240v = 0 \]
\[ 240v = 60 \]
\[ v = \frac{60}{240} = \frac{1}{4} = 0.25 \text{ m/s} \]

### Step 4: Determine the man's velocity relative to the ground
The man is walking forward at 1 m/s, while the trolley moves backward at 0.25 m/s. Therefore, the man's velocity relative to the ground (v_relative) is:
\[ v_{\text{relative}} = v_1 - v_2 \]
\[ v_{\text{relative}} = 1 - 0.25 = 0.75 \text{ m/s} \]

### Step 5: Calculate the displacement after 4 seconds
To find the displacement of the man relative to the ground after 4 seconds, we use the formula:
\[ \text{Displacement} = v_{\text{relative}} \cdot t \]
Where \( t = 4 \) seconds.
\[ \text{Displacement} = 0.75 \cdot 4 = 3 \text{ meters} \]

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

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