A boat going down stream crosses a float at a point A. `'t_1'` minutes later the boat reverses its direction and in the next `'t_2'` minutes it crosses the float at a distance L from the point A. The velocity of the river is 

Options

A. `2L//t_1`
B. `L//(t_1 + 2t_2)`
C. `2L//(2t_1 + t_2)`
D. `L//2t_1`

Answer

To find the velocity of the river, we can analyze the motion of the boat and the float in the context of the river's frame of reference. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- A boat crosses a float at point A while going downstream.
- After time \( t_1 \), the boat reverses direction.
- After an additional time \( t_2 \), the boat crosses the float again at a distance \( L \) from point A.

2. **Defining Variables**:
- Let \( v_0 \) be the velocity of the river.
- Let \( v_b \) be the speed of the boat relative to the water.
- The float moves with the velocity of the river, \( v_0 \).

3. **Distance Traveled by the River**:
- During the time \( t_1 \), the river carries the float downstream a distance of \( v_0 \cdot t_1 \).
- During the time \( t_2 \), the river carries the float downstream a distance of \( v_0 \cdot t_2 \).

4. **Total Distance Traveled**:
- The total distance traveled by the float from point A to the point where the boat crosses it again is \( L \).
- The boat travels downstream for \( t_1 \) and then upstream for \( t_2 \).
- Since the boat reverses direction, the effective distance traveled by the boat in the river frame is equal to the distance the float has moved downstream.

5. **Equating Distances**:
- The distance the boat travels downstream in time \( t_1 \) is \( (v_b + v_0) \cdot t_1 \).
- The distance the boat travels upstream in time \( t_2 \) is \( (v_b - v_0) \cdot t_2 \).
- The total distance covered by the float in both time intervals is:
\[ L = v_0 \cdot (t_1 + t_2) \]

6. **Using the Condition \( t_1 = t_2 \)**:
- Since \( t_1 = t_2 \), we can denote \( t_1 = t_2 = t \).
- Thus, the equation simplifies to:
\[ L = v_0 \cdot (t + t) = v_0 \cdot 2t \]

7. **Finding the Velocity of the River**:
- Rearranging the equation gives:
\[ v_0 = \frac{L}{2t} \]
- Therefore, the velocity of the river is:
\[ v_0 = \frac{L}{2t_1} \]

### Final Answer:
The velocity of the river \( v_0 \) is given by:
\[ v_0 = \frac{L}{2t_1} \]