A boat can fo across a lake and retyrb ub tune ` T_0` at a speed ` v`. On a rough day there is a uniform currednt at speed ` v_1` to help the onward hourney and impede the return jourmey. If the time taken to go across and return on the same day be (T) then ` T//T_0` is`

Options

A. (1 -v_1^2 //v^2` `1/((1 -v_1^2//v^2))`(1 + v_1^2/v^2)`
B. `1/((1 -v_1^2//v^2))`
C. `(1 + v_1^2/v^2)`
D. `1/ ((1 + v_1^2//v^2))`

Explanation

To solve the problem, we need to find the ratio \( \frac{T}{T_0} \) where \( T_0 \) is the time taken to cross a lake and return without any current, and \( T \) is the time taken to cross and return with a current affecting the boat's speed.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- The boat can cross a lake and return in time \( T_0 \) at speed \( v \).
- The distance across the lake is \( d \).
- The total distance for the round trip is \( 2d \).
- The time taken without current is given by \( T_0 = \frac{2d}{v} \).

2. **Effect of Current**:
- On the way to the other side, the current helps the boat, so the effective speed is \( v + v_1 \).
- On the return journey, the current opposes the boat, so the effective speed is \( v - v_1 \).

3. **Calculating Time with Current**:
- The time taken to go across the lake with the current is:
\[ T_1 = \frac{d}{v + v_1} \]
- The time taken to return against the current is:
\[ T_2 = \frac{d}{v - v_1} \]

4. **Total Time with Current**:
- The total time \( T \) for the round trip with current is:
\[ T = T_1 + T_2 = \frac{d}{v + v_1} + \frac{d}{v - v_1} \]

5. **Finding a Common Denominator**:
- To combine the two fractions, we find a common denominator:
\[ T = d \left( \frac{(v - v_1) + (v + v_1)}{(v + v_1)(v - v_1)} \right) \]
- Simplifying the numerator:
\[ T = d \left( \frac{2v}{v^2 - v_1^2} \right) \]

6. **Expressing Total Time**:
- Therefore, the total time \( T \) can be expressed as:
\[ T = \frac{2dv}{v^2 - v_1^2} \]

7. **Finding the Ratio \( \frac{T}{T_0} \)**:
- Now we can find the ratio \( \frac{T}{T_0} \):
\[ \frac{T}{T_0} = \frac{\frac{2dv}{v^2 - v_1^2}}{\frac{2d}{v}} = \frac{v}{v^2 - v_1^2} \]

8. **Final Expression**:
- Thus, the ratio \( \frac{T}{T_0} \) is:
\[ \frac{T}{T_0} = \frac{v}{v^2 - v_1^2} \]