A man can swim with speed u with respect to river. The widith of river is d. The speed of the river is zero at the banks and increses linearly to `V_(0)` till mid stream of the river then decreases to zero to the other bank. When man crosses the river in shoetest time, drift is equal to: `("given" u gt v_(0))`

Options

A. `(ud)/(v_(0))`
B. `(ud)/(2v_(0))`
C. `(v_(0)d)/(2u)`
D. `(v_(0)d)/(u)`

Explanation

To solve the problem of determining the drift of a man swimming across a river with a non-uniform current, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the River's Current**:
- The river has a width \( d \).
- The speed of the river increases linearly from \( 0 \) at the banks to \( V_0 \) at the midpoint, and then decreases back to \( 0 \) at the opposite bank.

2. **Swimming Speed**:
- The man swims with a speed \( u \) with respect to the river.

3. **Direction of Motion**:
- The man swims directly across the river (in the y-direction) to minimize the time taken to cross.

4. **Time to Cross the River**:
- The time \( t \) taken to cross the river can be calculated using the formula:
\[ t = \frac{d}{u} \]
- Here, \( d \) is the width of the river, and \( u \) is the man's swimming speed.

5. **Drift Calculation**:
- While the man swims across the river, he is also affected by the river's current.
- The velocity of the river at the midpoint (which is the maximum) is \( V_0 \).
- The drift \( x \) (displacement in the x-direction) can be calculated by considering the average velocity of the river during the time \( t \):
- The river's velocity increases linearly, so the average velocity \( V_{avg} \) over the width of the river is:
\[ V_{avg} = \frac{0 + V_0}{2} = \frac{V_0}{2} \]
- The drift \( x \) can then be calculated as:
\[ x = V_{avg} \cdot t = \frac{V_0}{2} \cdot \frac{d}{u} \]

6. **Final Expression for Drift**:
- Therefore, the drift \( x \) is given by:
\[ x = \frac{d V_0}{2u} \]

### Conclusion:
The drift when the man crosses the river in the shortest time is:
\[ \text{Drift} = \frac{d V_0}{2u} \]