A swimmer wishes to cross a 500m wide river flowing at a rate 5km/hr. His speed with respect to water is 3km/hr. (a) If the heads in a direction making an angle `theta` with the flow, he takes to cross the river. (b) Find the shortest possible time to cross the river.

Answer

To solve the problem step by step, we will break it down into parts as follows:

### Step 1: Understand the Problem
We have a river that is 500 meters wide, flowing at a speed of 5 km/hr. The swimmer has a speed of 3 km/hr relative to the water. The swimmer wants to cross the river at an angle θ with respect to the flow of the river.

### Step 2: Convert Speeds to SI Units
First, we need to convert the speeds from km/hr to m/s for consistency in units.
- Speed of the river (Vr) = 5 km/hr = \( \frac{5 \times 1000}{3600} \) m/s = \( \frac{5000}{3600} \) m/s = \( \frac{25}{18} \) m/s.
- Speed of the swimmer (Vs) = 3 km/hr = \( \frac{3 \times 1000}{3600} \) m/s = \( \frac{3000}{3600} \) m/s = \( \frac{5}{6} \) m/s.

### Step 3: Break Down the Swimmer's Velocity
The swimmer's velocity can be broken down into two components:
- \( V_{sy} = Vs \sin \theta \) (the component of the swimmer's speed in the direction across the river).
- \( V_{sx} = Vs \cos \theta \) (the component of the swimmer's speed in the direction of the river flow).

### Step 4: Calculate Time to Cross the River
The time taken to cross the river is determined by the vertical component of the swimmer's speed:
\[ t = \frac{d}{V_{sy}} = \frac{500 \text{ m}}{Vs \sin \theta} = \frac{500}{\frac{5}{6} \sin \theta} = \frac{600}{\sin \theta} \text{ seconds}. \]

### Step 5: Find the Shortest Possible Time
To minimize the time \( t \), we need to maximize \( \sin \theta \). The maximum value of \( \sin \theta \) is 1, which occurs when \( \theta = 90^\circ \).
- Therefore, the shortest time to cross the river occurs when:
\[ t_{min} = \frac{600}{1} = 600 \text{ seconds} = 10 \text{ minutes}. \]

### Final Answer
(a) The swimmer should head in a direction making an angle \( \theta \) with the flow, where \( \theta = 90^\circ \) for the shortest time.
(b) The shortest possible time to cross the river is 10 minutes.

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