The stream of a river is flowing with a speed of 2 km/h. A swimmer can swim at a speed of 4 km/h. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight?

Options

A. `90^(@)`
B. `60^(@)`
C. `120^(@)`
D. `150^(@)`

Explanation

To solve the problem of determining the direction a swimmer should take to cross a river straight while accounting for the river's current, we can follow these steps:

### Step 1: Understand the problem
The river is flowing with a speed of 2 km/h, and the swimmer can swim at a speed of 4 km/h. The swimmer wants to cross the river straight, meaning he wants to reach the point directly across from where he started, despite the current pushing him downstream.

**Hint:** Visualize the situation by drawing a diagram of the river, the swimmer's path, and the current.

### Step 2: Set up the vector components
Let the direction across the river (perpendicular to the flow) be the y-direction, and the direction of the river flow be the x-direction. The swimmer's velocity can be broken down into two components:
- \( V_{sy} \): the component of the swimmer's velocity in the y-direction (across the river).
- \( V_{sx} \): the component of the swimmer's velocity in the x-direction (against the river flow).

The swimmer's total speed is given as 4 km/h, so we can express this as:
\[ V_s = \sqrt{V_{sx}^2 + V_{sy}^2} \]

**Hint:** Remember that the swimmer's speed is the resultant of its components in both directions.

### Step 3: Relate the swimmer's speed and the river's speed
To cross the river straight, the swimmer must counteract the river's current. Therefore, the swimmer's x-component of velocity must equal the river's speed:
\[ V_{sx} = V_r = 2 \text{ km/h} \]

### Step 4: Calculate the y-component of the swimmer's velocity
Using the Pythagorean theorem, we can find the y-component of the swimmer's velocity:
\[ V_s^2 = V_{sx}^2 + V_{sy}^2 \]
Substituting the known values:
\[ 4^2 = 2^2 + V_{sy}^2 \]
\[ 16 = 4 + V_{sy}^2 \]
\[ V_{sy}^2 = 16 - 4 = 12 \]
\[ V_{sy} = \sqrt{12} = 2\sqrt{3} \text{ km/h} \]

**Hint:** Make sure to keep track of the units and the direction of each component.

### Step 5: Determine the angle of swimming
Now, we need to find the angle \( \theta \) that the swimmer should swim at with respect to the flow of the river. We can use the sine function to relate the components:
\[ \sin(\theta) = \frac{V_{sx}}{V_s} = \frac{2}{4} = \frac{1}{2} \]
Thus, we find:
\[ \theta = 30^\circ \]

**Hint:** Use trigonometric ratios to find the angle, and remember that the angle is measured from the direction of the river flow.

### Step 6: Determine the direction with respect to the river
Since the swimmer is swimming at an angle of \( 30^\circ \) upstream to counter the current, the angle with respect to the direction of the river flow is:
\[ 180^\circ - 30^\circ = 150^\circ \]

**Hint:** The angle you find should be measured from the direction of the river flow to ensure it is correct.

### Conclusion
The swimmer should swim at an angle of \( 30^\circ \) upstream from the direction of the river flow to cross the river straight.

### Final Answer
The direction of the swimmer with respect to the flow of the river should be \( 150^\circ \) from the downstream direction.

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