A swimmer crosses a flowing stream of width `omega` to and fro in time `t_(1)`. The time taken to cover the same distance up and down the stream is `t_(2)`. If `t_(3)` is the time the swimmer would take to swim a distance `2omega` in still water, then

Options

A. `t_(1)^(2)=t_(2)t_(3)`
B. `t_(2)^(2)=t_(1)t_(3)`
C. `t_(3)^(2)=t_(1)t_(2)`
D. `t_(3)=t_(1)+t_(2)`

Answer

To solve the problem step by step, we need to analyze the swimmer's movements in different scenarios and derive relationships between the times taken.

### Step 1: Understanding the Problem
The swimmer crosses a stream of width \( \omega \) in three different scenarios:
1. **To and fro across the stream** in time \( t_1 \).
2. **Upstream and downstream** across the same width in time \( t_2 \).
3. **Swimming a distance of \( 2\omega \)** in still water in time \( t_3 \).

### Step 2: Analyzing the First Condition (To and Fro)
When the swimmer crosses the stream to and fro, the total distance covered is \( 2\omega \). The effective velocity when swimming across the stream is given by:
\[ v_{resultant} = \sqrt{v_m^2 - v_r^2} \]
where \( v_m \) is the swimmer's speed in still water and \( v_r \) is the speed of the river.

The time taken \( t_1 \) can be expressed as:
\[ t_1 = \frac{2\omega}{\sqrt{v_m^2 - v_r^2}} \]

### Step 3: Analyzing the Second Condition (Upstream and Downstream)
In this case, the swimmer swims upstream and then downstream. The time taken \( t_2 \) can be calculated as:
\[ t_2 = \frac{\omega}{v_m - v_r} + \frac{\omega}{v_m + v_r} \]
This simplifies to:
\[ t_2 = \frac{2\omega}{v_m^2 - v_r^2} \cdot v_m \]

### Step 4: Analyzing the Third Condition (Still Water)
In still water, the swimmer swims a distance of \( 2\omega \). The time taken \( t_3 \) is:
\[ t_3 = \frac{2\omega}{v_m} \]

### Step 5: Establishing Relationships
Now we have expressions for \( t_1 \), \( t_2 \), and \( t_3 \):
1. \( t_1 = \frac{2\omega}{\sqrt{v_m^2 - v_r^2}} \)
2. \( t_2 = \frac{2\omega v_m}{v_m^2 - v_r^2} \)
3. \( t_3 = \frac{2\omega}{v_m} \)

### Step 6: Finding the Relation
We can now relate \( t_1 \), \( t_2 \), and \( t_3 \):
- Multiply \( t_2 \) and \( t_3 \):
\[ t_2 \cdot t_3 = \left(\frac{2\omega v_m}{v_m^2 - v_r^2}\right) \cdot \left(\frac{2\omega}{v_m}\right) = \frac{4\omega^2}{v_m^2 - v_r^2} \]
- Square \( t_1 \):
\[ t_1^2 = \left(\frac{2\omega}{\sqrt{v_m^2 - v_r^2}}\right)^2 = \frac{4\omega^2}{v_m^2 - v_r^2} \]

### Conclusion
From the above calculations, we can conclude that:
\[ t_1^2 = t_2 \cdot t_3 \]

### Final Answer
Thus, the relationship established is:
\[ t_1^2 = t_2 \cdot t_3 \]

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