The ratio of downstream drift of a person in crossing a river making same angles with downstream and upstream are respectively 2 : 1. The ratio of the speed of river u and the swimming speed of person v is:

Options

A. `(u)/(v)=1/2`
B. `(u)/(v)le 1/2`
C. `(u)/(v) ge 1/3`
D. `(u)/(v) le 3`

Explanation

To solve the problem, we need to analyze the situation involving a swimmer crossing a river and the downstream drift caused by the river's current. The swimmer makes the same angle with the current when swimming upstream and downstream, and we are given that the ratio of downstream drift to upstream drift is 2:1. Our goal is to find the ratio of the speed of the river (u) to the swimming speed of the person (v).

### Step-by-Step Solution:

1. **Understanding the Problem:**
- Let the swimmer make an angle θ with the river current while swimming upstream and downstream.
- The downstream drift (x2) is twice the upstream drift (x1), so we have the ratio:
\[ \frac{x2}{x1} = \frac{2}{1} \]

2. **Setting Up the Components:**
- When swimming upstream, the swimmer's velocity can be broken down into components:
- Velocity against the current: \( v \cos \theta \)
- Velocity across the river: \( v \sin \theta \)

3. **Calculating Time Taken to Cross the River:**
- Let the width of the river be \( d \).
- The time taken to cross the river while swimming upstream is:
\[ t_1 = \frac{d}{v \sin \theta} \]

4. **Calculating Upstream Drift (x1):**
- The effective velocity of the swimmer against the river current is \( v \cos \theta - u \).
- The upstream drift (x1) can be calculated as:
\[ x1 = (u - v \cos \theta) t_1 = (u - v \cos \theta) \left(\frac{d}{v \sin \theta}\right) \]

5. **Calculating Downstream Drift (x2):**
- The effective velocity of the swimmer with the river current is \( v \cos \theta + u \).
- The downstream drift (x2) can be calculated as:
\[ x2 = (u + v \cos \theta) t_2 = (u + v \cos \theta) \left(\frac{d}{v \sin \theta}\right) \]

6. **Setting Up the Ratio:**
- Now we can set up the ratio of the drifts:
\[ \frac{x2}{x1} = \frac{(u + v \cos \theta)}{(u - v \cos \theta)} = \frac{2}{1} \]

7. **Cross-Multiplying to Solve for u:**
- Cross-multiplying gives us:
\[ 1 \cdot (u + v \cos \theta) = 2 \cdot (u - v \cos \theta) \]
- Expanding this:
\[ u + v \cos \theta = 2u - 2v \cos \theta \]
- Rearranging terms:
\[ u + 2v \cos \theta = 2u \]
- This simplifies to:
\[ u = 3v \cos \theta \]

8. **Finding the Ratio of Speeds:**
- Now, we can find the ratio of the speed of the river to the swimming speed:
\[ \frac{u}{v} = 3 \cos \theta \]

9. **Conclusion:**
- Since \( \cos \theta \) can take values between 0 and 1, the ratio \( \frac{u}{v} \) can be at most 3. Therefore, the final answer is:
\[ \frac{u}{v} \leq 3 \]