A river is `20 m` wide. River speed is `3 m//s.` A boat starts with velocity `2sqrt2 m//s` at angle `45^@` from the river current (relative to river)
(a) Find the time taken by the boat to reach the opposite bank.
(b) How far from the point directly opposite to the starting point does the boat reach the opposite bank?

Explanation

To solve the problem step by step, we will break it down into two parts: (a) finding the time taken by the boat to reach the opposite bank and (b) finding how far from the point directly opposite to the starting point the boat reaches the opposite bank.

### Part (a): Time Taken by the Boat to Reach the Opposite Bank

1. **Identify the width of the river (W)**:
The river is given to be 20 meters wide.

2. **Determine the velocity of the boat (V_boat)**:
The boat has a velocity of \(2\sqrt{2} \, \text{m/s}\) at an angle of \(45^\circ\) relative to the river current.

3. **Calculate the vertical component of the boat's velocity (V_AB)**:
Since the boat is moving at an angle of \(45^\circ\), we can find the vertical component using:
\[ V_{AB} = V_{\text{boat}} \cdot \cos(45^\circ) = 2\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 2 \, \text{m/s} \]

4. **Calculate the time taken to cross the river (t)**:
The time taken to reach the opposite bank can be calculated using the formula:
\[ t = \frac{W}{V_{AB}} = \frac{20 \, \text{m}}{2 \, \text{m/s}} = 10 \, \text{s} \]

### Part (b): How Far from the Point Directly Opposite the Starting Point the Boat Reaches

1. **Determine the river's speed (V_river)**:
The speed of the river is given as \(3 \, \text{m/s}\).

2. **Calculate the horizontal component of the boat's velocity (V_BC)**:
The horizontal component of the boat's velocity can be calculated as:
\[ V_{BC} = V_{\text{river}} + V_{\text{boat}} \cdot \sin(45^\circ) = 3 \, \text{m/s} + 2\sqrt{2} \cdot \frac{1}{\sqrt{2}} = 3 \, \text{m/s} + 2 \, \text{m/s} = 5 \, \text{m/s} \]

3. **Calculate the drift (D)**:
The drift can be calculated using the formula:
\[ D = V_{BC} \cdot t = 5 \, \text{m/s} \cdot 10 \, \text{s} = 50 \, \text{m} \]

### Final Answers:
(a) The time taken by the boat to reach the opposite bank is **10 seconds**.
(b) The boat reaches the opposite bank **50 meters** downstream from the point directly opposite to the starting point.