A beg (mass M) hangs by a long thred and a bullet (mass m) comes horizontally with velocity v and gets caught in the bag. The for the combined (bag+bullet) system -

Options

A. `"Momentum is "(mvM)/((M+m))`
B. `"KE is "(mv^(2))/2`
C. `"Momentum is "(mv(M+m))/M`
D. `"KE is "(m^(2)v^(2))/(2(M+m))`

Explanation

To solve the problem, we need to analyze the situation using the principles of conservation of momentum and kinetic energy. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the System
We have a bag of mass \( M \) hanging by a thread and a bullet of mass \( m \) coming horizontally with a velocity \( v \). When the bullet gets caught in the bag, we need to analyze the combined system of the bag and the bullet.

### Step 2: Apply Conservation of Momentum
Before the collision, only the bullet has momentum. The initial momentum \( p_1 \) of the system is given by:
\[ p_1 = m \cdot v \]
After the bullet gets caught in the bag, the total mass of the system becomes \( M + m \). Let \( V \) be the final velocity of the combined system after the bullet is caught. By conservation of momentum:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
Thus, we have:
\[ m \cdot v = (M + m) \cdot V \]

### Step 3: Solve for Final Velocity \( V \)
Rearranging the equation from Step 2 to find \( V \):
\[ V = \frac{m \cdot v}{M + m} \]

### Step 4: Calculate Final Kinetic Energy
The initial kinetic energy \( KE_{initial} \) of the bullet is:
\[ KE_{initial} = \frac{1}{2} m v^2 \]
After the bullet is caught in the bag, the kinetic energy \( KE_{final} \) of the combined system is:
\[ KE_{final} = \frac{1}{2} (M + m) V^2 \]
Substituting \( V \) from Step 3:
\[ KE_{final} = \frac{1}{2} (M + m) \left(\frac{m \cdot v}{M + m}\right)^2 \]
This simplifies to:
\[ KE_{final} = \frac{1}{2} (M + m) \cdot \frac{m^2 v^2}{(M + m)^2} = \frac{m^2 v^2}{2(M + m)} \]

### Step 5: Conclusion
From the calculations, we have derived the final velocity \( V \) and the final kinetic energy \( KE_{final} \) of the combined system.

### Summary of Results
- Final Velocity \( V = \frac{m \cdot v}{M + m} \)
- Final Kinetic Energy \( KE_{final} = \frac{m^2 v^2}{2(M + m)} \)