A bomb of mass `m` at rest at the coordinate origin explodes into three equal pieces. At a certain instant one piece is on the x-axis at `x=40cm` and another is at `x=20cm, y=-60cm`. The position of the third piece is

Options

A. `x=60 cm,y=60 cm`
B. `x=-60 cm,y=-60 cm`
C. `x=-60cm,y=60 cm`
D. `x=60 cm,y=-60 cm`

Explanation

To solve the problem, we need to find the position of the third piece of the bomb after it explodes into three equal pieces. We will use the principle of conservation of momentum and the concept of the center of mass.

### Step-by-Step Solution:

1. **Understand the System**:
- The bomb of mass `m` is initially at rest at the origin (0,0).
- After the explosion, it breaks into three equal pieces, each of mass \( \frac{m}{3} \).

2. **Identify the Positions of the Known Pieces**:
- The first piece is at \( (40 \text{ cm}, 0) \).
- The second piece is at \( (20 \text{ cm}, -60 \text{ cm}) \).
- Let the position of the third piece be \( (x, y) \).

3. **Calculate the Center of Mass Before and After the Explosion**:
- Initially, the center of mass (CM) is at the origin:
\[ x_{CM, initial} = 0, \quad y_{CM, initial} = 0 \]
- After the explosion, the center of mass must remain at the origin since no external forces are acting on the system:
\[ x_{CM, final} = 0, \quad y_{CM, final} = 0 \]

4. **Set Up the Equations for the Center of Mass**:
- The formula for the center of mass in the x-direction:
\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} \]
- For our case:
\[ 0 = \frac{\frac{m}{3} \cdot 40 + \frac{m}{3} \cdot 20 + \frac{m}{3} \cdot x}{m} \]
- Simplifying:
\[ 0 = \frac{40 + 20 + x}{3} \]
- Thus:
\[ 40 + 20 + x = 0 \implies x = -60 \text{ cm} \]

5. **Repeat for the y-direction**:
- The formula for the center of mass in the y-direction:
\[ 0 = \frac{\frac{m}{3} \cdot 0 + \frac{m}{3} \cdot (-60) + \frac{m}{3} \cdot y}{m} \]
- Simplifying:
\[ 0 = \frac{-60 + y}{3} \]
- Thus:
\[ -60 + y = 0 \implies y = 60 \text{ cm} \]

6. **Final Position of the Third Piece**:
- The position of the third piece is \( (-60 \text{ cm}, 60 \text{ cm}) \).

### Conclusion:
The position of the third piece after the explosion is \( (-60 \text{ cm}, 60 \text{ cm}) \).