On a horizontal frictionless frozen lake, a girl (36 kg) and a box (9 kg) are connected to each other by means of a rope. Initially they are 20 m apart. The girl exerts a horizontal force on the box, pulling it towards her. How far has the girl traveled when she meets the box `gt`

Options

A. 10 m
B. Since there is no friction, the girl will not move
C. 16 m
D. 4m

Answer

To solve the problem, we need to determine how far the girl has traveled when she meets the box. We can use the concept of the center of mass (COM) to find this distance.

### Step-by-Step Solution:

1. **Identify the Masses and Initial Distance**:
- Mass of the girl, \( m_g = 36 \, \text{kg} \)
- Mass of the box, \( m_b = 9 \, \text{kg} \)
- Initial distance between the girl and the box, \( d = 20 \, \text{m} \)

2. **Calculate the Position of the Center of Mass**:
The center of mass \( x_{cm} \) of the system can be calculated using the formula:
\[ x_{cm} = \frac{m_g \cdot x_g + m_b \cdot x_b}{m_g + m_b} \]
Let's assume the girl starts at position \( x_g = 0 \) and the box starts at position \( x_b = 20 \, \text{m} \).

Substituting the values:
\[ x_{cm} = \frac{36 \cdot 0 + 9 \cdot 20}{36 + 9} = \frac{180}{45} = 4 \, \text{m} \]
This means the center of mass is located 4 meters from the girl’s starting position.

3. **Determine the Distances Traveled**:
Let \( x \) be the distance the girl travels towards the box. When they meet, the girl will have traveled \( x \) meters, and the box will have traveled \( 20 - x \) meters.

4. **Set Up the Equation for the Center of Mass**:
Since the center of mass does not move, we can set up the equation based on the distances traveled:
\[ m_g \cdot (x + 0) + m_b \cdot (20 - x) = m_g \cdot 4 + m_b \cdot 4 \]
Simplifying this gives:
\[ 36x + 9(20 - x) = 4(36 + 9) \]
\[ 36x + 180 - 9x = 180 \]
\[ 27x = 0 \]
\[ x = 4 \, \text{m} \]

5. **Conclusion**:
The girl travels 4 meters towards the box when they meet.

### Final Answer:
The girl has traveled **4 meters** when she meets the box.