A bullet of `10 g` leaves the barrel of gun with a velocity of `600 m/s`. If the barrel of gun is `50cm` long and mass of gun is `3kg`, then value of impulse supplied to the gun will be :

Options

A. `6 Ns `
B. `12 Ns`
C. `36 Ns`
D. `3 Ns`

Answer

To solve the problem, we need to determine the impulse supplied to the gun when the bullet is fired. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We have a bullet of mass \( m_b = 10 \, \text{g} = 0.01 \, \text{kg} \) that leaves the barrel of a gun with a velocity \( v_b = 600 \, \text{m/s} \). The mass of the gun \( m_g = 3 \, \text{kg} \). We need to find the impulse supplied to the gun.

### Step 2: Apply the Law of Conservation of Momentum
According to the law of conservation of momentum, the total momentum before firing is equal to the total momentum after firing. Initially, both the bullet and the gun are at rest, so the initial momentum is zero.

After the bullet is fired, the momentum of the bullet and the gun must balance out:
\[ 0 = m_b \cdot v_b + m_g \cdot (-v_g) \]
Where \( v_g \) is the recoil velocity of the gun.

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ m_b \cdot v_b = m_g \cdot v_g \]
Substituting the known values:
\[ 0.01 \cdot 600 = 3 \cdot v_g \]
Calculating the left side:
\[ 6 = 3 \cdot v_g \]

### Step 4: Solve for \( v_g \)
Now, solve for \( v_g \):
\[ v_g = \frac{6}{3} = 2 \, \text{m/s} \]

### Step 5: Calculate the Impulse
Impulse is defined as the change in momentum. The momentum of the gun after firing is given by:
\[ \text{Impulse} = m_g \cdot v_g \]
Substituting the values:
\[ \text{Impulse} = 3 \cdot 2 = 6 \, \text{Ns} \]

### Final Answer
The value of impulse supplied to the gun is \( 6 \, \text{Ns} \).

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