A bullet of mass 10 g is fired from a gun of mass 1 kg with recoil velocity of gun 5 m/s. The muzzle velocity will be

Options

A. 30 km/min
B. 60 km/min
C. 30 m/s
D. 500 m/s

Explanation

To solve the problem of finding the muzzle velocity of a bullet fired from a gun, we can use the principle of conservation of momentum. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the System
We have a bullet of mass \( m = 10 \, \text{g} = 0.01 \, \text{kg} \) and a gun of mass \( M = 1 \, \text{kg} \). The gun recoils with a velocity \( V_g = 5 \, \text{m/s} \) in the opposite direction when the bullet is fired.

### Step 2: Apply Conservation of Momentum
According to the law of conservation of momentum, the total momentum before firing must equal the total momentum after firing. Initially, both the bullet and the gun are at rest, so the initial momentum is:
\[ \text{Initial Momentum} = 0 \]

After the bullet is fired, the final momentum of the system (bullet + gun) can be expressed as:
\[ \text{Final Momentum} = (m \cdot V) + (M \cdot -V_g) \]
Where:
- \( V \) is the muzzle velocity of the bullet (which we need to find).
- The negative sign for \( V_g \) indicates that the gun recoils in the opposite direction to the bullet.

### Step 3: Set Up the Equation
Setting the initial momentum equal to the final momentum gives us:
\[ 0 = (m \cdot V) + (M \cdot -V_g) \]
Rearranging this equation, we have:
\[ m \cdot V = M \cdot V_g \]

### Step 4: Solve for Muzzle Velocity \( V \)
Now, we can solve for \( V \):
\[ V = \frac{M \cdot V_g}{m} \]
Substituting the known values:
- \( M = 1 \, \text{kg} \)
- \( V_g = 5 \, \text{m/s} \)
- \( m = 0.01 \, \text{kg} \)

We get:
\[ V = \frac{1 \, \text{kg} \cdot 5 \, \text{m/s}}{0.01 \, \text{kg}} = \frac{5}{0.01} = 500 \, \text{m/s} \]

### Step 5: Conclusion
The muzzle velocity of the bullet is \( 500 \, \text{m/s} \).

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