A shell of mass 0.01 kg fired by a gun of mass 10 kg. If the muzzle speed of the shell is `50 ms^(-1)`, what is the recoil speed of the gun?

Options

A. 0.01 m/s
B. 0.02 m/s
C. 0.03 m/s
D. 0.05 m/s

Answer

To solve the problem of finding the recoil speed of the gun when a shell is fired, we can use the principle of conservation of momentum. Here are the steps to arrive at the solution:

### Step 1: Understand the Conservation of Momentum
The law of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In this case, the system consists of the gun and the shell.

### Step 2: Set Up the Equation
Before the shell is fired, both the gun and the shell are at rest, so the total initial momentum is zero. After the shell is fired, the momentum of the shell and the gun must still add up to zero.

Let:
- Mass of the shell, \( m_s = 0.01 \, \text{kg} \)
- Mass of the gun, \( m_g = 10 \, \text{kg} \)
- Muzzle speed of the shell, \( v_s = 50 \, \text{m/s} \)
- Recoil speed of the gun, \( v_g \) (which we need to find)

According to the conservation of momentum:
\[ m_s \cdot v_s + m_g \cdot v_g = 0 \]

### Step 3: Substitute the Known Values
Substituting the known values into the equation:
\[ 0.01 \cdot 50 + 10 \cdot v_g = 0 \]

### Step 4: Calculate the Momentum of the Shell
Calculating the momentum of the shell:
\[ 0.5 + 10 \cdot v_g = 0 \]

### Step 5: Solve for the Recoil Speed of the Gun
Rearranging the equation to solve for \( v_g \):
\[ 10 \cdot v_g = -0.5 \]
\[ v_g = \frac{-0.5}{10} = -0.05 \, \text{m/s} \]

### Step 6: Interpret the Result
The negative sign indicates that the gun recoils in the opposite direction to the shell. Therefore, the recoil speed of the gun is \( 0.05 \, \text{m/s} \).

### Final Answer
The recoil speed of the gun is \( 0.05 \, \text{m/s} \).

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