A river is flowing towards with a velocity of `5 m s^-1`. The boat velocity is `10 ms^-1`. The boat crosses the river by shortest path. Hence,

Options

A. The direction of boat's velocity is `30^@` west of north.
B. The direction of boat's velocity is north - west.
C. Resultant velocity is `5 sqrt(3) ms^-1`.
D. Resultant velocity of boat is `5 sqrt(2) m s^-1`.

Explanation

To solve the problem of a boat crossing a river with a current, we can break down the steps as follows:

### Step 1: Understand the Problem
The river flows with a velocity \( V_r = 5 \, \text{m/s} \) and the boat has a velocity \( V_b = 10 \, \text{m/s} \). The boat is crossing the river by the shortest path, which means it is moving perpendicular to the riverbank.

### Step 2: Visualize the Motion
We can visualize the situation as a right triangle where:
- The horizontal leg represents the velocity of the river (\( V_r \)).
- The vertical leg represents the component of the boat's velocity that is perpendicular to the riverbank.
- The hypotenuse represents the boat's actual velocity (\( V_b \)).

### Step 3: Set Up the Relationship
Since the boat is crossing the river by the shortest path, it must have a component of its velocity that counters the river's current. We can use trigonometric functions to relate these velocities.

### Step 4: Use the Sine Function
Using the sine function, we can express the relationship as follows:
\[ \sin(\theta) = \frac{V_r}{V_b} \]
Substituting the values:
\[ \sin(\theta) = \frac{5}{10} = \frac{1}{2} \]

### Step 5: Calculate the Angle
To find the angle \( \theta \):
\[ \theta = \arcsin\left(\frac{1}{2}\right) = 30^\circ \]
This angle is measured from the direction of the river's flow towards the direction of the boat's motion, which is 30 degrees west of north.

### Step 6: Find the Resultant Velocity
To find the resultant velocity of the boat with respect to the river, we can use the Pythagorean theorem:
\[ V_{resultant} = \sqrt{V_b^2 - V_r^2} \]
Substituting the values:
\[ V_{resultant} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3} \, \text{m/s} \]

### Conclusion
The boat crosses the river at an angle of 30 degrees west of north, and its resultant velocity with respect to the river is \( 5\sqrt{3} \, \text{m/s} \).

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