A man of mass m stands on a long flat car of mass M , moving with velocity V.If he now begins to run with velocity u, with respect to the car , in the same direction as V, the the velocity of the car will be

Options

A. (a)`V - m u//M`
B. (b)`V - m u//(m + M)`
C. (c)`V + m u//(m + M)`
D. (d)`V - ut(M - m)//(M + m)`

Explanation

To solve the problem, we will use the principle of conservation of momentum. Let's break down the steps:

### Step-by-Step Solution:

1. **Identify the Initial Momentum:**
The initial momentum of the system (man + car) can be expressed as:
\[ P_{\text{initial}} = (M + m)V \]
where \(M\) is the mass of the car, \(m\) is the mass of the man, and \(V\) is the initial velocity of the car.

2. **Define the Final Velocities:**
Let \(v_c\) be the final velocity of the car after the man starts running, and \(v_m\) be the final velocity of the man. Since the man runs with velocity \(u\) with respect to the car, we can express the man's velocity as:
\[ v_m = v_c + u \]

3. **Write the Final Momentum:**
The final momentum of the system after the man starts running is:
\[ P_{\text{final}} = Mv_c + mv_m = Mv_c + m(v_c + u) = Mv_c + mv_c + mu \]
Simplifying this gives:
\[ P_{\text{final}} = (M + m)v_c + mu \]

4. **Apply Conservation of Momentum:**
According to the conservation of momentum, the initial momentum must equal the final momentum:
\[ (M + m)V = (M + m)v_c + mu \]

5. **Rearranging the Equation:**
Rearranging the equation to solve for \(v_c\):
\[ (M + m)v_c = (M + m)V - mu \]
\[ v_c = \frac{(M + m)V - mu}{M + m} \]

6. **Final Expression for the Velocity of the Car:**
Thus, the final velocity of the car \(v_c\) is:
\[ v_c = V - \frac{mu}{M + m} \]

### Conclusion:
The velocity of the car after the man starts running is given by:
\[ v_c = V - \frac{mu}{M + m} \]