A `10g` bullet is short from a `5kg` gun with a velocity of `400m//s`. What is the speed of recoil of the gun?

Options

A. -0.8 m/s
B. 80 m/s
C. 0.4 m/s
D. 20 m/s

Explanation

To solve the problem of finding the speed of recoil of the gun when a bullet is fired, we will use the law of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the Law of Conservation of Momentum
The law states that in the absence of external forces, the total momentum before an event must equal the total momentum after the event.

### Step 2: Identify Initial Momentum
Before the bullet is fired, both the gun and the bullet are at rest. Therefore, the initial momentum is:
\[ \text{Initial Momentum} = 0 \]

### Step 3: Identify Final Momentum
After the bullet is fired, the bullet and the gun will have momentum. The momentum of the bullet can be calculated using the formula:
\[ \text{Momentum} = \text{mass} \times \text{velocity} \]
- Mass of the bullet = 10 g = \(10 \times 10^{-3}\) kg = \(0.01\) kg
- Velocity of the bullet = 400 m/s

So, the momentum of the bullet is:
\[ \text{Momentum of bullet} = 0.01 \, \text{kg} \times 400 \, \text{m/s} = 4 \, \text{kg m/s} \]

Let \(v\) be the recoil velocity of the gun. The mass of the gun is 5 kg. Therefore, the momentum of the gun is:
\[ \text{Momentum of gun} = 5 \, \text{kg} \times v \]

### Step 4: Set Up the Equation
According to the conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
Thus, we have:
\[ 0 = \text{Momentum of bullet} + \text{Momentum of gun} \]
This can be written as:
\[ 0 = 4 \, \text{kg m/s} + (5 \, \text{kg} \times v) \]

### Step 5: Solve for \(v\)
Rearranging the equation gives:
\[ 5v = -4 \]
Now, divide both sides by 5:
\[ v = -\frac{4}{5} \]
Calculating this gives:
\[ v = -0.8 \, \text{m/s} \]

### Step 6: Interpret the Result
The negative sign indicates that the direction of the gun's recoil is opposite to the direction of the bullet's motion.

### Final Answer
The speed of recoil of the gun is \(0.8 \, \text{m/s}\) in the opposite direction to the bullet.

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