A bullet of mass 0.01 kg is fired from a gun weighing 5.0 kg. If the initial speed of the bullet is 250 m/s, calculate the speed with which the gun recoils.

Options

A. `-0.50`m/s
B. `-0.25`m/s
C. `+0.05` m/s
D. `+0.25` m/s

Explanation

To solve the problem of calculating the recoil speed of a gun when a bullet is fired, we can use the principle of conservation of momentum. Here's a step-by-step solution:

### Step 1: Understand the system
We have a bullet of mass \( m = 0.01 \, \text{kg} \) and a gun of mass \( M = 5.0 \, \text{kg} \). The bullet is fired with an initial speed \( v = 250 \, \text{m/s} \).

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

\[ \text{Initial momentum} = 0 \]

After the bullet is fired, the momentum of the bullet and the momentum of the gun must balance each other. Let \( V' \) be the recoil speed of the gun (which will be in the opposite direction to the bullet's motion). The final momentum can be expressed as:

\[ \text{Final momentum} = m \cdot v + M \cdot (-V') \]

Setting the initial momentum equal to the final momentum gives us:

\[ 0 = m \cdot v - M \cdot V' \]

### Step 3: Rearranging the equation
Rearranging the equation to solve for the recoil speed \( V' \):

\[ M \cdot V' = m \cdot v \]

\[ V' = \frac{m \cdot v}{M} \]

### Step 4: Substitute the values
Now we can substitute the known values into the equation:

\[ V' = \frac{0.01 \, \text{kg} \cdot 250 \, \text{m/s}}{5.0 \, \text{kg}} \]

### Step 5: Calculate the recoil speed
Calculating the above expression:

\[ V' = \frac{2.5 \, \text{kg m/s}}{5.0 \, \text{kg}} = 0.5 \, \text{m/s} \]

### Step 6: Determine the direction
Since the gun recoils in the opposite direction of the bullet, we denote the recoil speed as negative:

\[ V' = -0.5 \, \text{m/s} \]

### Final Answer
The speed with which the gun recoils is \( -0.5 \, \text{m/s} \) (indicating the opposite direction to the bullet).

---