A shell at rest at the origin explodes into three fragments of masses 1 kg, 2 kg and m kg. Thel kg and 2 kg pieces fly off with speeds of 12 m/s along X-axis and 16 m/s along y-axis respectively. If the m kg piece flies off with a speed of 40 m/s, the total mass of the shell must be

Options

A. 3.8 kg
B. 4kg
C. 4.5 kg
D. 5 kg

Explanation

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.

### Step-by-Step Solution:

1. **Identify the Initial Momentum:**
The shell is at rest before the explosion, so the initial momentum \( P_{\text{initial}} = 0 \).

2. **Identify the Masses and Velocities:**
- Mass of the first fragment \( m_1 = 1 \, \text{kg} \) with velocity \( v_1 = 12 \, \text{m/s} \) along the x-axis.
- Mass of the second fragment \( m_2 = 2 \, \text{kg} \) with velocity \( v_2 = 16 \, \text{m/s} \) along the y-axis.
- Mass of the third fragment \( m_3 = m \, \text{kg} \) with velocity \( v_3 = 40 \, \text{m/s} \) (direction is unknown).

3. **Calculate the Momentum of Each Fragment:**
- Momentum of the first fragment:
\[ P_1 = m_1 \cdot v_1 = 1 \, \text{kg} \cdot 12 \, \text{m/s} = 12 \, \text{kg m/s} \, \text{(along x-axis)} \]
- Momentum of the second fragment:
\[ P_2 = m_2 \cdot v_2 = 2 \, \text{kg} \cdot 16 \, \text{m/s} = 32 \, \text{kg m/s} \, \text{(along y-axis)} \]
- Momentum of the third fragment:
\[ P_3 = m_3 \cdot v_3 = m \cdot 40 \, \text{m/s} \]

4. **Set Up the Conservation of Momentum Equation:**
Since the initial momentum is zero, the total momentum after the explosion must also equal zero:
\[ P_1 + P_2 + P_3 = 0 \]
This can be expressed as:
\[ 12 \hat{i} + 32 \hat{j} + (m \cdot 40) \hat{v} = 0 \]

5. **Calculate the Magnitude of the Total Momentum:**
The total momentum in vector form is:
\[ P_{\text{total}} = \sqrt{(12)^2 + (32)^2} = \sqrt{144 + 1024} = \sqrt{1168} \approx 34.17 \, \text{kg m/s} \]

6. **Relate the Momentum of the Third Fragment:**
The momentum of the third fragment must balance the momentum of the first two fragments:
\[ |P_3| = 34.17 \, \text{kg m/s} \]
Thus,
\[ m \cdot 40 = 34.17 \]

7. **Solve for the Mass \( m \):**
\[ m = \frac{34.17}{40} \approx 0.85425 \, \text{kg} \]

8. **Calculate the Total Mass of the Shell:**
The total mass of the shell is:
\[ \text{Total mass} = m_1 + m_2 + m_3 = 1 + 2 + 0.85425 \approx 3.85425 \, \text{kg} \]
Rounding this to two decimal places gives approximately \( 3.85 \, \text{kg} \).

### Final Answer:
The total mass of the shell must be approximately **3.85 kg**.