Assertion:- The magnitude of velocity of two boats relative to river is same. Both boats start simultaneously from same point on one bank They may reach opposite bank simultaneously moving along different straight line paths.
Reason:- For boats to cross the river in same time, the compnonet of their velocity relative to river in direction normal to flow should be same.

Options

A. If both Assertion `&` Reason are True `&` the Reason is a correct explanation of the Assertion.
B. If both Assertion `&` Reason are True but Reason is not a correct explanation of the Assertion.
C. If Assertion is True but the Reason is False.
D. If both Assertion `&` Reason are False.

Answer

To solve the problem, we need to analyze the assertion and the reason given in the question.

### Step 1: Understanding the Assertion
The assertion states that two boats can reach the opposite bank of a river simultaneously while moving along different straight paths, provided their velocities relative to the river are the same.

### Step 2: Analyzing the Velocity Components
When the boats are crossing the river, their velocity can be broken down into two components:
- The component of velocity in the direction across the river (normal to the flow of the river).
- The component of velocity in the direction of the river's flow.

### Step 3: Condition for Simultaneous Crossing
For both boats to reach the opposite bank at the same time, the component of their velocities that is directed across the river must be the same. This means that even if they are moving at different angles, as long as their vertical (y-axis) components of velocity are equal, they will take the same time to cross.

### Step 4: Mathematical Representation
Let:
- \( v \) = magnitude of the velocity of each boat relative to the river.
- \( \theta_1 \) and \( \theta_2 \) = angles made by the boats with respect to the riverbank.

The vertical component of the velocity for each boat can be expressed as:
- For Boat 1: \( v_{1y} = v \cdot \sin(\theta_1) \)
- For Boat 2: \( v_{2y} = v \cdot \sin(\theta_2) \)

For both boats to cross the river in the same time \( T \):
\[ v \cdot \sin(\theta_1) = v \cdot \sin(\theta_2) \]

### Step 5: Conclusion
Since the magnitudes of their velocities relative to the river are the same, and the components of their velocities in the direction normal to the flow of the river are also equal, both boats can indeed reach the opposite bank simultaneously despite following different paths.

### Final Answer
Both the assertion and reason are true, and the reason correctly explains the assertion.

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