A man can row a boat in still water with a velocity of `8 kmph`.Water is flowing in a river with a velocity of `4 kmph`. At what angle should he row the boat so as to reach the exact opposite point

Options

A. `150^(@)` to flow of water.
B. `120^(@)` to flow of water.
C. `30^(@)` to flow of water.
D. `90^(@)` to flow of water.

Explanation

To solve the problem of how a man should row a boat in a river to reach the exact opposite point, we can follow these steps:

### Step 1: Understand the Problem
The man rows a boat in still water with a velocity of \( V_m = 8 \) km/h. The river flows with a velocity of \( V_r = 4 \) km/h. We need to find the angle \( \alpha \) at which he should row the boat to reach the point directly opposite to his starting point.

### Step 2: Set Up the Coordinate System
Let’s establish a coordinate system:
- The direction of the river flow (to the right) is the positive x-direction.
- The direction perpendicular to the river flow (upwards) is the positive y-direction.

### Step 3: Identify the Velocities
- The velocity of the man relative to the water is \( V_m = 8 \) km/h at an angle \( \alpha \) with respect to the upstream direction.
- The velocity of the river is \( V_r = 4 \) km/h in the positive x-direction.

### Step 4: Resolve the Man's Velocity
The man's velocity can be resolved into two components:
- The x-component (horizontal): \( V_{mx} = V_m \cos(\alpha) \)
- The y-component (vertical): \( V_{my} = V_m \sin(\alpha) \)

### Step 5: Set Up the Condition for Reaching the Opposite Point
To reach the exact opposite point, the horizontal component of the man's velocity must equal the velocity of the river:
\[ V_{mx} = V_r \]
Substituting the expressions:
\[ V_m \cos(\alpha) = V_r \]
\[ 8 \cos(\alpha) = 4 \]

### Step 6: Solve for \( \cos(\alpha) \)
Rearranging the equation gives:
\[ \cos(\alpha) = \frac{4}{8} = \frac{1}{2} \]

### Step 7: Find the Angle \( \alpha \)
Now, we can find \( \alpha \):
\[ \alpha = \cos^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \alpha = 60^\circ \]

### Step 8: Determine the Direction Relative to the Flow of Water
Since the angle is measured upstream, the angle to the flow of the water is:
\[ 180^\circ - 60^\circ = 120^\circ \]

### Final Answer
The man should row the boat at an angle of \( 120^\circ \) to the flow of the water to reach the exact opposite point.

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