A bullet of mass `m` and velocity `v` is fired into a large block of mass `M`. The final velocity of the system is

Options

A. `(M)/(m+M).v`
B. `(m)/(m+M).v`
C. `(m+M)/(m).v`
D. None of these

Explanation

To solve the problem of finding the final velocity of a system consisting of a bullet fired into a large block, 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 \( m \) moving with a velocity \( v \), and a large block of mass \( M \) that is initially at rest.

### Step 2: Identify Initial Momentum
Before the bullet hits the block, the initial momentum \( P_i \) of the system can be calculated as:
\[ P_i = \text{momentum of bullet} + \text{momentum of block} \]
Since the block is at rest, its momentum is zero. Thus,
\[ P_i = mv + 0 = mv \]

### Step 3: Identify Final Momentum
After the bullet embeds itself into the block, they move together as a single system. The total mass of this combined system is:
\[ m' = M + m \]
Let the final velocity of the system after the collision be \( v' \). The final momentum \( P_f \) is then:
\[ P_f = (M + m)v' \]

### Step 4: Apply Conservation of Momentum
According to the law of conservation of momentum, the initial momentum must equal the final momentum:
\[ P_i = P_f \]
Substituting the expressions we derived:
\[ mv = (M + m)v' \]

### Step 5: Solve for Final Velocity
To find the final velocity \( v' \), we can rearrange the equation:
\[ v' = \frac{mv}{M + m} \]

### Conclusion
Thus, the final velocity of the system after the bullet is fired into the block is:
\[ v' = \frac{mv}{M + m} \]