A continuous stream of particles, of mass `m` and velocity `r`, is emitted from a source at a rate of `n` per second. The particles travel along a straight line, collide with a body of mass `M` and get embedded in the body. If the mass `M` was originally at rest, its velocity when it has received `N` particles will be

Options

A. `(mvn)/(Nm+n)`
B. `(mvN)/(Nm+M)`
C. `(mv)/(Nm+M)`
D. `(Nm+M)/(mv)`

Explanation

To solve the problem, we will use the principle of conservation of momentum. The key steps are as follows:

### Step 1: Understand the System
We have a body of mass \( M \) initially at rest, and a continuous stream of particles, each of mass \( m \) and velocity \( r \), colliding with the body and getting embedded in it.

### Step 2: Initial Momentum Calculation
Since the body of mass \( M \) is at rest, its initial momentum is:
\[ P_{\text{initial}} = 0 \]
The momentum of the incoming particles can be calculated as follows:
If \( n \) particles are emitted per second, the momentum contributed by one particle is:
\[ P_{\text{particle}} = m \cdot r \]
Thus, for \( n \) particles, the total initial momentum of the particles is:
\[ P_{\text{initial particles}} = n \cdot (m \cdot r) = nmr \]

### Step 3: Final Momentum Calculation
After \( N \) particles collide with the body and get embedded in it, the total mass of the system becomes:
\[ M + Nm \]
Let the final velocity of the combined mass after \( N \) particles have been embedded be \( V \). The final momentum of the system is:
\[ P_{\text{final}} = (M + Nm) \cdot V \]

### Step 4: Apply Conservation of Momentum
According to the conservation of momentum, the total initial momentum must equal the total final momentum:
\[ P_{\text{initial particles}} = P_{\text{final}} \]
Substituting the values we calculated:
\[ nmr = (M + Nm) \cdot V \]

### Step 5: Solve for Final Velocity \( V \)
Rearranging the equation to solve for \( V \):
\[ V = \frac{nmr}{M + Nm} \]

### Final Result
Thus, the velocity of the body after it has received \( N \) particles is:
\[ V = \frac{nmr}{M + Nm} \]