A shell of mass 200g is fired by a gun of mass 100kg. If the muzzle speed of the shell is `80ms^(-1)`, then the recoil speed of the gun is

Options

A. `16cms^(-1)`
B. `8cms^(-1)`
C. `8ms^(-1)`
D. `16ms^(-1)`

Explanation

To solve the problem, we will use the principle of conservation of momentum. The total momentum before firing the shell must be equal to the total momentum after firing the shell.

### Step-by-Step Solution:

1. **Identify the masses and speeds:**
- Mass of the shell (m_shell) = 200 g = 0.2 kg (since 1 g = 0.001 kg)
- Mass of the gun (m_gun) = 100 kg
- Muzzle speed of the shell (v_shell) = 80 m/s
- Recoil speed of the gun (v_gun) = ?

2. **Initial momentum of the system:**
- Before the shell is fired, both the gun and the shell are at rest. Therefore, the initial momentum (P_initial) of the system is:
\[ P_{\text{initial}} = 0 \, \text{(since both are at rest)} \]

3. **Final momentum of the system:**
- After the shell is fired, the momentum of the shell and the gun can be expressed as:
\[ P_{\text{final}} = m_{\text{shell}} \cdot v_{\text{shell}} + m_{\text{gun}} \cdot v_{\text{gun}} \]
- Since the gun recoils in the opposite direction to the shell, we can consider the direction of the shell's velocity as positive and the gun's velocity as negative.

4. **Apply the conservation of momentum:**
- According to the conservation of momentum:
\[ P_{\text{initial}} = P_{\text{final}} \]
- Therefore:
\[ 0 = (0.2 \, \text{kg} \cdot 80 \, \text{m/s}) + (100 \, \text{kg} \cdot v_{\text{gun}}) \]

5. **Solve for the recoil speed of the gun (v_gun):**
- Rearranging the equation gives:
\[ 100 \, \text{kg} \cdot v_{\text{gun}} = - (0.2 \, \text{kg} \cdot 80 \, \text{m/s}) \]
- Calculate the right side:
\[ 100 \, \text{kg} \cdot v_{\text{gun}} = -16 \, \text{kg m/s} \]
- Now, divide both sides by 100 kg:
\[ v_{\text{gun}} = \frac{-16 \, \text{kg m/s}}{100 \, \text{kg}} = -0.16 \, \text{m/s} \]

6. **Convert to centimeters per second:**
- To convert the speed from meters per second to centimeters per second:
\[ v_{\text{gun}} = -0.16 \, \text{m/s} \times 100 = -16 \, \text{cm/s} \]
- The negative sign indicates that the gun recoils in the opposite direction to the shell.

### Final Answer:
The recoil speed of the gun is **16 cm/s** in the opposite direction to the shell.

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