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. (a)The direction of boat’s velocity is `30^(@)` west of north
B. (b)The direction of boat’s velocity is north-west
C. (c)Resultant velocity is `5sqrt(3) ms^(-1)`
D. (d)Resultant velocity is `5sqrt(2) ms^(-1)`

Explanation

To solve the problem of a boat crossing a river with a current, we will analyze the situation step by step.

### Step 1: Understand the Given Information
- Velocity of the river (Vr) = 5 m/s (flowing towards the east)
- Velocity of the boat (Vb) = 10 m/s (the boat crosses the river)

### Step 2: Determine the Shortest Path
To cross the river by the shortest path, the boat must head upstream at an angle such that its resultant velocity allows it to reach the opposite bank directly across from its starting point.

### Step 3: Set Up the Diagram
We can visualize the situation with a right triangle:
- The vertical side represents the width of the river (let's denote it as 'd').
- The horizontal side represents the velocity of the river (Vr = 5 m/s).
- The hypotenuse represents the velocity of the boat (Vb = 10 m/s).

### Step 4: Apply Trigonometry
Using trigonometric relationships, we can find the angle (θ) at which the boat must head to cross directly:
- The sine of the angle θ can be expressed as:
\[ \sin(\theta) = \frac{Vr}{Vb} = \frac{5}{10} = \frac{1}{2} \]
- From this, we find:
\[ \theta = 30^\circ \]

### Step 5: Determine the Direction of the Boat
Since the river flows east and the boat heads upstream, the angle θ is measured from the north towards the west. Thus, the direction of the boat is:
- 30° west of north.

### Step 6: Calculate the Resultant Velocity of the Boat
To find the resultant velocity of the boat relative to the ground, we can use the Pythagorean theorem:
\[ V_r = \sqrt{Vb^2 - Vr^2} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} = 5\sqrt{3} \, \text{m/s} \]

### Step 7: Conclusion
Based on the calculations:
- The direction of the boat is 30° west of north.
- The resultant velocity of the boat relative to the ground is \(5\sqrt{3}\) m/s.

### Final Answer
- The correct options are:
- Direction of the boat: 30° west of north (Option A is correct).
- Resultant velocity: \(5\sqrt{3}\) m/s (Option C is correct).