A plank of mass 2m and length L is al rest on a frictionless horizontal floor At one end of it, a child of mass m is standing as shown in figure If child walks and reaches the other and then the distance travelled by child wrt ground will be

Options

A. `L /3`
B. `L /2`
C. `2L /3`
D. `3L /4`

Answer

To solve the problem, we need to find the distance traveled by the child with respect to the ground when he walks from one end of the plank to the other. We will use the concept of the center of mass and the principle of conservation of momentum.

### Step-by-Step Solution:

1. **Identify the System**:
- We have a plank of mass \(2m\) and length \(L\) on a frictionless surface.
- A child of mass \(m\) is standing on one end of the plank.

2. **Initial Center of Mass Calculation**:
- The initial position of the center of mass (CM) of the system can be calculated using the formula:
\[ x_{CM} = \frac{m \cdot 0 + 2m \cdot \frac{L}{2}}{m + 2m} = \frac{0 + mL}{3m} = \frac{L}{3} \]
- Here, the child is at position \(0\) and the plank’s center of mass is at \(\frac{L}{2}\).

3. **Final Center of Mass Calculation**:
- When the child walks to the other end of the plank, we need to find the new position of the center of mass. Let’s denote the distance the child walks as \(x\).
- The final position of the child will be \(L\) (the other end of the plank).
- The new position of the center of mass will be:
\[ x'_{CM} = \frac{m \cdot L + 2m \cdot \frac{L}{2}}{m + 2m} = \frac{mL + mL}{3m} = \frac{2L}{3} \]

4. **Conservation of Center of Mass**:
- Since there are no external forces acting on the system, the center of mass of the system must remain constant. Therefore, the initial center of mass position must equal the final center of mass position:
\[ \frac{L}{3} = \frac{2L}{3} - d \]
- Here, \(d\) is the distance the plank moves in the opposite direction due to the child walking.

5. **Finding the Distance Traveled by the Child**:
- The distance the child travels with respect to the ground can be calculated as:
\[ \text{Distance}_{\text{child}} = L - d \]
- From the conservation of center of mass, we can find \(d\):
\[ d = \frac{L}{3} \]
- Therefore, the distance traveled by the child with respect to the ground is:
\[ \text{Distance}_{\text{child}} = L - \frac{L}{3} = \frac{2L}{3} \]

### Final Answer:
The distance traveled by the child with respect to the ground is \(\frac{2L}{3}\).