An explosion blows a rock into three parts. Two pieces go off at right angles to each other. 1.00 kg piece with a velocity 12m/sec and the other 2.00 kg piece with a velocity 8m/sec. If the third piece flies off with a velocity 40m/sec, then the mass of the third piece is -

Options

A. 0.2 kg
B. 0.3 kg
C. 0.4 kg
D. 0.5 kg

Answer

To solve the problem, we will use the principle of conservation of momentum. The total momentum before the explosion must equal the total momentum after the explosion. Since the rock is initially at rest, the total initial momentum is zero.

### Step-by-Step Solution:

1. **Identify the masses and velocities of the pieces:**
- Mass of piece 1 (m1) = 1.00 kg, velocity (v1) = 12 m/s (along the x-axis)
- Mass of piece 2 (m2) = 2.00 kg, velocity (v2) = 8 m/s (along the y-axis)
- Mass of piece 3 (m3) = ?, velocity (v3) = 40 m/s (unknown direction)

2. **Set up the momentum equations:**
- The total momentum in the x-direction before the explosion is 0.
- The total momentum in the y-direction before the explosion is also 0.

3. **Calculate the momentum of the first two pieces:**
- Momentum of piece 1 in the x-direction: \( p_{1x} = m_1 \cdot v_1 = 1.00 \, \text{kg} \cdot 12 \, \text{m/s} = 12 \, \text{kg m/s} \)
- Momentum of piece 2 in the y-direction: \( p_{2y} = m_2 \cdot v_2 = 2.00 \, \text{kg} \cdot 8 \, \text{m/s} = 16 \, \text{kg m/s} \)

4. **Express the momentum of the third piece:**
- Let the mass of the third piece be \( m_3 \).
- The momentum of the third piece can be expressed as \( p_{3} = m_3 \cdot v_3 \).
- Since the third piece is moving at 40 m/s, we can express its momentum in terms of its components. However, since we don't know the direction, we will consider the total momentum.

5. **Apply the conservation of momentum:**
- In the x-direction:
\[ 0 = 12 + m_3 \cdot v_{3x} \]
- In the y-direction:
\[ 0 = 16 + m_3 \cdot v_{3y} \]

6. **Calculate the magnitude of the momentum of the third piece:**
- The magnitude of the momentum of the third piece must balance the total momentum of the first two pieces:
\[ m_3 \cdot 40 = \sqrt{(12)^2 + (16)^2} \]
- Calculate \( \sqrt{(12)^2 + (16)^2} \):
\[ \sqrt{144 + 256} = \sqrt{400} = 20 \]

7. **Set up the equation:**
\[ m_3 \cdot 40 = 20 \]

8. **Solve for \( m_3 \):**
\[ m_3 = \frac{20}{40} = 0.5 \, \text{kg} \]

### Final Answer:
The mass of the third piece is **0.5 kg**.