A grenade having mass of 10 kg flying horizontally with a velocity of 10 m/s explodes into two fragments. The larger fragment has a velocity of 25 m/s in the direction of the intial velocity of the grenade. The smaller fragment has a velocity of 12.5 m/s in the opposite dirction. The masses of the fragments are

Options

A. 2 kg , 8 kg
B. 3 kg, 7kg
C. 4 kg, 6 kg
D. `4.5 kg, 5.5 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.

### Step-by-Step Solution:

1. **Calculate Initial Momentum:**
The initial momentum \( p_{\text{initial}} \) of the grenade can be calculated using the formula:
\[ p_{\text{initial}} = m \cdot v \]
where \( m = 10 \, \text{kg} \) (mass of the grenade) and \( v = 10 \, \text{m/s} \) (initial velocity).
\[ p_{\text{initial}} = 10 \, \text{kg} \cdot 10 \, \text{m/s} = 100 \, \text{kg m/s} \]

2. **Define Variables for Fragments:**
Let \( m_1 \) be the mass of the larger fragment and \( m_2 \) be the mass of the smaller fragment. According to the problem:
- The larger fragment moves with a velocity \( v_1 = 25 \, \text{m/s} \) (in the same direction as the grenade).
- The smaller fragment moves with a velocity \( v_2 = -12.5 \, \text{m/s} \) (in the opposite direction).

3. **Write the Equation for Final Momentum:**
The final momentum \( p_{\text{final}} \) after the explosion can be expressed as:
\[ p_{\text{final}} = m_1 \cdot v_1 + m_2 \cdot v_2 \]
Substituting the values of \( v_1 \) and \( v_2 \):
\[ p_{\text{final}} = m_1 \cdot 25 + m_2 \cdot (-12.5) \]
\[ p_{\text{final}} = 25m_1 - 12.5m_2 \]

4. **Set Initial Momentum Equal to Final Momentum:**
By conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
Thus, we have:
\[ 100 = 25m_1 - 12.5m_2 \quad \text{(Equation 1)} \]

5. **Use the Conservation of Mass:**
The total mass before and after the explosion must also be equal:
\[ m_1 + m_2 = 10 \quad \text{(Equation 2)} \]

6. **Solve the Equations:**
From Equation 2, we can express \( m_2 \) in terms of \( m_1 \):
\[ m_2 = 10 - m_1 \]
Substitute \( m_2 \) into Equation 1:
\[ 100 = 25m_1 - 12.5(10 - m_1) \]
Simplifying this:
\[ 100 = 25m_1 - 125 + 12.5m_1 \]
\[ 100 + 125 = 37.5m_1 \]
\[ 225 = 37.5m_1 \]
\[ m_1 = \frac{225}{37.5} = 6 \, \text{kg} \]

7. **Find \( m_2 \):**
Now substitute \( m_1 \) back into Equation 2:
\[ m_2 = 10 - m_1 = 10 - 6 = 4 \, \text{kg} \]

### Final Answer:
The masses of the fragments are:
- Larger fragment \( m_1 = 6 \, \text{kg} \)
- Smaller fragment \( m_2 = 4 \, \text{kg} \)