Width of a river is `30 m,` velocity is ` 2 m//s` and rowing velocity is `5 m//s` at `37^@` from the direction of river current (a) find the time taken to cross the river, (b) drift of the boatman while reaching the other shore.

Explanation

To solve the problem step by step, we will break it down into two parts as per the question: (a) finding the time taken to cross the river and (b) calculating the drift of the boatman while reaching the other shore.

### Step-by-Step Solution:

**Given:**
- Width of the river (D) = 30 m
- Velocity of the river (Vr) = 2 m/s
- Rowing velocity of the boat (Vb) = 5 m/s
- Angle of rowing with respect to the river current (θ) = 37°

### Part (a): Time Taken to Cross the River

1. **Calculate the vertical component of the rowing velocity:**
The component of the rowing velocity in the direction perpendicular to the river (towards the opposite shore) can be calculated using the sine function:
\[ V_{BR} = V_b \cdot \sin(θ) = 5 \cdot \sin(37°) \]
Using the value of \(\sin(37°) \approx \frac{3}{5}\):
\[ V_{BR} = 5 \cdot \frac{3}{5} = 3 \text{ m/s} \]

2. **Calculate the time taken to cross the river:**
The time taken (t) to cross the river can be calculated using the formula:
\[ t = \frac{D}{V_{BR}} = \frac{30}{3} = 10 \text{ seconds} \]

### Part (b): Drift of the Boatman While Reaching the Other Shore

1. **Calculate the horizontal component of the rowing velocity:**
The horizontal component of the rowing velocity (in the direction of the river current) can be calculated using the cosine function:
\[ V_{BH} = V_b \cdot \cos(θ) = 5 \cdot \cos(37°) \]
Using the value of \(\cos(37°) \approx \frac{4}{5}\):
\[ V_{BH} = 5 \cdot \frac{4}{5} = 4 \text{ m/s} \]

2. **Calculate the effective velocity in the direction of the river:**
The effective velocity in the direction of the river (downstream) is the sum of the horizontal component of the rowing velocity and the velocity of the river:
\[ V_{effective} = V_{BH} + V_r = 4 + 2 = 6 \text{ m/s} \]

3. **Calculate the drift distance:**
The drift (D_drift) while crossing the river can be calculated using the formula:
\[ D_{drift} = V_{effective} \cdot t = 6 \cdot 10 = 60 \text{ meters} \]

### Final Answers:
- (a) Time taken to cross the river = **10 seconds**
- (b) Drift of the boatman while reaching the other shore = **60 meters**