A swimmer wants to cross a river from point A to point B. Line AB makes an angle of 30$^\circ$ with the flow of river. Magnitude of velocity of the swimmer is same as that of the river. The angle $\theta$ with the line AB should be ________ $^\circ$, so that the swimmer reaches point B.
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In relative velocity problems, drawing a vector triangle ($\vec{v}_{resultant} = \vec{v}_{relative} + \vec{v}_{frame}$) is often the clearest way to solve. Applying the Law of Sines or Cosines to this triangle can simplify the calculations.
To solve the problem of determining the angle θ such that the swimmer reaches point B:
The swimmer's velocity is the same as the velocity of the river. This implies that the swimmer must swim in a specific direction to counteract the river's flow and reach point B, which is directly across from point A.
Given that line AB makes an angle of 30° with the river's flow, and using vector components, the swimmer needs to swim at an angle θ relative to line AB such that the component of the swimmer's velocity perpendicular to the river's flow cancels out the component along the flow.
The problem setup can be visualized by breaking the swimmer's velocity into components: one perpendicular to AB and one along AB. For the swimmer to reach point B, the resultant vector should aim in the direction of AB.
The geometry implies that the angle θ with the horizontal direction (flow direction) covers the vectors properly.
Using trigonometry and the concept of relative velocity:
Since the magnitudes of the swimmer's velocity and the river's velocity are equal, and considering the 30° inclination:
cosθ = cos30°
This results in θ = 30°.
Conclusion: The angle θ should be 30°, which fits within the given range of 30 to 30°. Therefore, the swimmer should swim at this angle relative to AB to reach point B successfully.
