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. `V-m u//M`
B. `V-m u //(m+M)`
C. `V+m u//(m+m)`
D. `V-u(M-m)//(M+m)`

Answer

To find the velocity of the car after the man starts running, we can use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the Initial Conditions
- The man has a mass \( m \) and is standing on a car of mass \( M \).
- The car is moving with an initial velocity \( V \).

### Step 2: Define the Final Conditions
- When the man runs with a velocity \( u \) relative to the car in the same direction as the car's motion, we need to find the new velocity of the car, which we will denote as \( V' \).

### Step 3: Apply the Conservation of Momentum
Since there are no external horizontal forces acting on the system (the man and the car), the total momentum before the man starts running must equal the total momentum after he starts running.

- **Initial Momentum**:
\[ P_{initial} = (M + m) \cdot V \]

- **Final Momentum**:
After the man runs, he moves with a velocity \( V' + u \) (since \( u \) is the velocity of the man relative to the car). Therefore, the final momentum is:
\[ P_{final} = m \cdot (V' + u) + M \cdot V' \]

### Step 4: Set Up the Equation
Setting the initial momentum equal to the final momentum:
\[ (M + m) \cdot V = m \cdot (V' + u) + M \cdot V' \]

### Step 5: Expand and Rearrange the Equation
Expanding the right side:
\[ (M + m) \cdot V = mV' + mu + MV' \]
Combining like terms:
\[ (M + m) \cdot V = (M + m)V' \]

### Step 6: Solve for \( V' \)
Now, we can isolate \( V' \):
\[ (M + m)V' = (M + m)V - mu \]
Dividing both sides by \( (M + m) \):
\[ V' = V - \frac{mu}{M + m} \]

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