A river is flowing with a speed of \(1 \mathrm{~km} / \mathrm{hr}\). \(\mathrm{A}\) swimmer wants to go to point 'C' starting from 'A'. He swims with a speed of \(5 \mathrm{~km} / \mathrm{hr}\), at an angle \(\theta\) w.r.t. the river. If \(A B=B C=400 \mathrm{~m}\). Then the value of \(\theta\) is: (1) \(37^{\circ}\) (2) \(30^{\circ}\) (3) \(60^{\circ}\) (4) \(53^{\circ}\)

In physics, the concept of relative velocity is used to determine the velocity of an object as observed from a different frame of reference. This is particularly important in problems involving multiple moving objects, like a swimmer in a river. The swimmer's velocity relative to the riverbank can be found by combining the swimmer's velocity relative to the water and the velocity of the river current itself.
To calculate relative velocity, we use vector addition. Imagine a swimmer who swims with a velocity vector \(\textbf{v}_{\text{sw}}\) in still water. However, the river is flowing with its own velocity vector \(\textbf{v}_{\text{r}}\). The effective velocity of the swimmer as seen from the riverbank is the vector sum of these two velocities:
\[\textbf{v}_{\text{net}} = \textbf{v}_{\text{sw}} + \textbf{v}_{\text{r}}\]
Understanding this can help solve related physics problems by resolving the velocities involved into their respective components.

Trigonometry is widely used in physics to solve problems involving angles and distances. When breaking down velocity components or calculating necessary directions, trigonometric functions such as sine, cosine, and tangent are essential tools.
In our swimmer problem, we use trigonometry to determine the correct angle \(\theta\) the swimmer must swim at to reach his destination directly across the river. By employing the tangent function, we can relate the angle to the distances involved:
\[\tan(\theta) = \frac{\text{Downstream Displacement}}{\text{Distance AB}}\]
Hence, if a swimmer needs to counteract the flow of the river, the tangent of the angle \(\theta\) is the ratio of how far the swimmer is displaced downstream (due to river current) to the distance he swims perpendicular to the river flow. This relationship helps find the required direction the swimmer must take.

Problems involving a river current, such as the swimmer scenario, require an understanding of the interplay between the swimmer's speed and the current's speed. These problems often involve calculating resultant velocities and understanding how the current affects movement across the river.
Key steps to solve such problems include:

• Determining the swimmer's speed relative to still water.
• Identifying the speed and direction of the river current.
• Calculating the effective displacement caused by the current.
• Using trigonometric relationships to find the necessary swimming direction.

For example, if a swimmer has to reach a point directly across a river, he must swim at an angle that counters the downstream flow caused by the current. The current pushes the swimmer downstream, so he must aim upstream at a calculated angle based on the flows involved. By understanding these principles, students can effectively address and solve river current problems with more confidence.