A bullet of mass P is fired with velocity Q in a large body of mass R. The final velocity of the system will be

Options

A. `R/P+R`
B. `PQ/P+R`
C. `(P+ Q)/R`
D. `(P+R)/P Q`

Answer

To solve the problem of finding the final velocity of a system when a bullet of mass \( P \) is fired with velocity \( Q \) into a large body of mass \( R \), we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the System
We have two objects:
- A bullet with mass \( P \) and initial velocity \( Q \).
- A large body with mass \( R \) which is initially at rest (velocity = 0).

### Step 2: Write the Initial Momentum
The initial momentum of the system can be calculated as follows:
- Momentum of the bullet = \( P \times Q \)
- Momentum of the large body = \( R \times 0 = 0 \)

So, the total initial momentum \( p_{\text{initial}} \) is:
\[ p_{\text{initial}} = P \times Q + 0 = P \times Q \]

### Step 3: Write the Final Momentum
After the bullet is fired, both the bullet and the large body will move together with a final velocity \( V \). The total final momentum \( p_{\text{final}} \) is:
\[ p_{\text{final}} = (P + R) \times V \]

### Step 4: Apply Conservation of Momentum
According to the law of conservation of momentum, the total initial momentum is equal to the total final momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
Substituting the expressions we derived:
\[ P \times Q = (P + R) \times V \]

### Step 5: Solve for Final Velocity \( V \)
To find \( V \), we rearrange the equation:
\[ V = \frac{P \times Q}{P + R} \]

### Conclusion
Thus, the final velocity \( V \) of the system after the bullet is fired into the large body is:
\[ V = \frac{P \times Q}{P + R} \]