Relative Velocity In River

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A swimmer can swim at 1.5 m/s in still water. River flows east at 0.5 m/s. To go straight north, at what angle upstream (from north) should the swimmer aim?

RankGuru · Accepted Answer

1. **Vector Diagram**: We want the resultant velocity vector ($\vec{v}_r$) of the swimmer relative to ground to be straight North. This means the East-West component of swimmer's velocity relative to water must cancel the river's speed.
2. **Identify Components**: Let $v_s = 1.5$ m/s (speed in still water) and $v_{river} = 0.5$ m/s (East).
3. **Aim Direction**: If the swimmer aims at an angle $	heta$ West of North:
   - Swimmer East-West component $= v_s \sin 	heta$ (Westward).
4. **Condition for Straight North**: $v_s \sin 	heta = v_{river}$:
   - $1.5 \sin 	heta = 0.5 \implies \sin 	heta = \frac{0.5}{1.5} = \frac{1}{3} \approx 0.333$.
5. **Calculate Angle**: $	heta = \arcsin(0.333) \approx 19.47^\circ$.
6. **Conclusion**: To cross straight, the swimmer should aim approximately **$19^\circ$ West of North**.

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Relative Velocity In River

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