An open topped rail road car of mass M has an initial velocity `v_(0)` along a straight horizontal frictionless track. It suddenly starts raising at timet = 0. The rain drops fall vertically with velocity u and add a mass m kg/sec of water. The velocity of car after t second will be (assuming that it is not completely filled with water)

Options

A. `v_(0)+m(u)/(M)`
B. `(Mv_(0))/(M+mt)`
C. `(Mv_(0)+ut)/(M+ut)`
D. `v_(0)+(m""ut)/(M+ut)`

Explanation

To solve the problem, we will apply the principle of conservation of momentum. Here are the steps to derive the final velocity of the railroad car after time \( t \):

### Step 1: Understand the system
- The railroad car has an initial mass \( M \) and an initial velocity \( v_0 \).
- It starts raining at \( t = 0 \), adding mass to the car at a rate of \( m \) kg/sec.
- The raindrops fall vertically with velocity \( u \), which does not affect the horizontal motion of the car.

### Step 2: Identify the initial momentum
- The initial momentum \( p_i \) of the system (car) is given by:
\[ p_i = M \cdot v_0 \]

### Step 3: Determine the mass after time \( t \)
- After \( t \) seconds, the mass of the car will increase due to the rain. The additional mass added is:
\[ \text{Added mass} = m \cdot t \]
- Therefore, the total mass of the car after \( t \) seconds becomes:
\[ M_f = M + m \cdot t \]

### Step 4: Apply conservation of momentum
- Since there are no external horizontal forces acting on the system, the horizontal momentum is conserved. Thus, the final momentum \( p_f \) of the system is:
\[ p_f = M_f \cdot v_f \]
where \( v_f \) is the final velocity of the car after time \( t \).

### Step 5: Set initial momentum equal to final momentum
- According to the conservation of momentum:
\[ p_i = p_f \]
Substituting the expressions for momentum:
\[ M \cdot v_0 = (M + m \cdot t) \cdot v_f \]

### Step 6: Solve for the final velocity \( v_f \)
- Rearranging the equation to solve for \( v_f \):
\[ v_f = \frac{M \cdot v_0}{M + m \cdot t} \]

### Conclusion
- The final velocity of the railroad car after time \( t \) is given by:
\[ v_f = \frac{M \cdot v_0}{M + m \cdot t} \]