A shell of mass 0.020 kg is fired by a gun of mass 100 kg. If the speed of the shell is `80 m/s`, the recoil speed of the gun would be

Options

A. 14 m/s
B. 0.012 m/s
C. 0.016 m/s
D. 100 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’s a step-by-step breakdown of the solution:

### Step 1: Understand the System
We have a gun and a shell. When the shell is fired, the gun will experience a recoil in the opposite direction. We need to apply the conservation of momentum principle to this system.

### Step 2: Set Up the Conservation of Momentum Equation
Initially, both the gun and the shell are at rest, so the total initial momentum (PI) is zero. After the shell is fired, the total final momentum (PF) must also equal zero, according to the conservation of momentum.

Let:
- Mass of the shell (m_shell) = 0.020 kg
- Speed of the shell (v_shell) = 80 m/s
- Mass of the gun (m_gun) = 100 kg
- Recoil speed of the gun (v_gun) = ?

The conservation of momentum can be expressed as:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
\[ 0 = (m_{\text{shell}} \cdot v_{\text{shell}}) + (m_{\text{gun}} \cdot v_{\text{gun}}) \]

### Step 3: Substitute the Known Values
Substituting the known values into the equation gives:
\[ 0 = (0.020 \, \text{kg} \cdot 80 \, \text{m/s}) + (100 \, \text{kg} \cdot v_{\text{gun}}) \]

### Step 4: Solve for the Recoil Speed of the Gun
Rearranging the equation to solve for \( v_{\text{gun}} \):
\[ 100 \, v_{\text{gun}} = - (0.020 \cdot 80) \]
\[ 100 \, v_{\text{gun}} = -1.6 \]
\[ v_{\text{gun}} = -\frac{1.6}{100} \]
\[ v_{\text{gun}} = -0.016 \, \text{m/s} \]

### Step 5: Interpret the Result
The negative sign indicates that the direction of the gun's recoil is opposite to the direction of the shell's motion. Thus, the magnitude of the recoil speed of the gun is:
\[ v_{\text{gun}} = 0.016 \, \text{m/s} \]

### Final Answer
The recoil speed of the gun is **0.016 m/s**.

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