A gun of mass 3 kg fires a bullet of mass 30 g. The bullet takes 0.003s to move through the barrel of the gun and acquires a velocity of 100 m/s. Calculate:
(i) the velocity with which the gun recoils
(ii) the force exerted on gunman due to recoil of the gun.

Answer

To solve the problem step by step, we will use the principles of conservation of momentum and the formula for force.

### Given Data:
- Mass of the gun (M_g) = 3 kg
- Mass of the bullet (M_b) = 30 g = 0.030 kg (since 1 g = 0.001 kg)
- Velocity of the bullet (V_b) = 100 m/s
- Time taken by the bullet to move through the barrel (t) = 0.003 s

### Step 1: Calculate the Recoil Velocity of the Gun
According to the law of conservation of momentum, the total momentum before firing is equal to the total momentum after firing.

**Initial momentum (P_initial)** = 0 (since both the gun and bullet are at rest)

**Final momentum (P_final)** = Momentum of the gun + Momentum of the bullet
- Momentum of the gun = M_g * V_g (where V_g is the recoil velocity of the gun)
- Momentum of the bullet = M_b * V_b

Setting the initial momentum equal to the final momentum:
\[ P_{initial} = P_{final} \]
\[ 0 = M_g \cdot V_g + M_b \cdot V_b \]

Substituting the values:
\[ 0 = 3 \cdot V_g + 0.030 \cdot 100 \]
\[ 0 = 3 \cdot V_g + 3 \]

Rearranging the equation to solve for V_g:
\[ 3 \cdot V_g = -3 \]
\[ V_g = -1 \text{ m/s} \]

The negative sign indicates that the gun recoils in the opposite direction to the bullet.

### Step 2: Calculate the Force Exerted on the Gunman Due to Recoil
The force exerted can be calculated using the formula:
\[ F = \frac{\Delta P}{\Delta t} \]
where \( \Delta P \) is the change in momentum and \( \Delta t \) is the time interval.

**Change in momentum of the gun (ΔP)**:
- Initial momentum of the gun = 0
- Final momentum of the gun = M_g * V_g = 3 kg * (-1 m/s) = -3 kg·m/s

Thus, the change in momentum (ΔP) is:
\[ \Delta P = P_{final} - P_{initial} = -3 - 0 = -3 \text{ kg·m/s} \]

Now substituting into the force formula:
\[ F = \frac{-3}{0.003} \]
\[ F = -1000 \text{ N} \]

The negative sign indicates that the force is in the opposite direction of the bullet's motion.

### Final Answers:
(i) The velocity with which the gun recoils is **1 m/s** (in the opposite direction of the bullet).
(ii) The force exerted on the gunman due to recoil of the gun is **1000 N**.

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