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. `(mnN)/(Nm+M)`
B. `(mvN)/(NM+M)`
C. `(mv)/(Nm+M)`
D. `(Nm+M)/(NM)`

Explanation

To solve the problem, we will use the principle of conservation of momentum. Here's the step-by-step solution:

### Step 1: Understand the system
We have a continuous stream of particles, each with mass `m` and traveling at velocity `r`. These particles collide with a stationary body of mass `M` and get embedded in it. We need to find the velocity of the combined mass after `N` particles have been embedded.

### Step 2: Calculate the total mass after `N` particles are embedded
When `N` particles, each of mass `m`, collide with the body of mass `M`, the total mass of the system after the collision will be:
\[ \text{Total mass} = M + N \cdot m \]

### Step 3: Apply the conservation of momentum
Before the collision, the momentum of the system is due to the particles only, as the body of mass `M` is at rest. The momentum of `N` particles is given by:
\[ \text{Initial momentum} = N \cdot m \cdot r \]
After the collision, the total momentum of the combined mass (body + embedded particles) moving with velocity `V` is:
\[ \text{Final momentum} = (M + N \cdot m) \cdot V \]
According to the conservation of momentum:
\[ \text{Initial momentum} = \text{Final momentum} \]
Thus, we can write:
\[ N \cdot m \cdot r = (M + N \cdot m) \cdot V \]

### Step 4: Solve for the final velocity `V`
Rearranging the equation to solve for `V` gives:
\[ V = \frac{N \cdot m \cdot r}{M + N \cdot m} \]

### Conclusion
The velocity of the body after it has received `N` particles is:
\[ V = \frac{N \cdot m \cdot r}{M + N \cdot m} \]