A swimmer wants to cross a river from point A to point B. Line AB makes an angle of 30° with the flow of river. Magnitude of velocity of the swimmer is same as that of the river. The angle θ with the line AB should be _________°, so that the swimmer reaches point B.

Solution

Physics sir · 4 steps

Equal speeds mean the resultant bisects the angle, so that bisector must lie along AB.

1. Set up directions. Let both speeds be v. The river flows along the bank, which is 30° below AB. The swimmer heads at θ on the other side of AB, so his direction makes 30° + θ with the flow.
2. Cancel sideways drift. To reach B, the ground velocity must point along AB, so the velocity components perpendicular to AB must cancel. The river pushes him off AB by v sin 30°, and his own swimming pulls him back by v sin θ.

   v sin θ = v sin 30° ⇒ θ = 30°

3. Bisector check. Two vectors of equal size add along the line that bisects the angle between them. The angle between swimmer and river is 60°, and its bisector lies 30° from the flow, which is exactly AB.
4. So. The swimmer must head at θ = 30° to AB, which is 60° to the flow.

The common slip: 60. Reporting the swimmer's angle with the river flow (30° + θ) gives 60, but the question asks for the angle with AB.

The idea to keep. To move along a chosen line, cancel the velocity components perpendicular to that line. With equal speeds, the resultant always bisects the angle between the two velocities.

Answer from the NTA final answer key. Solution by PrepWiser.