Relative velocity is the velocity of one object as observed from another. For objects A and B: v_{A/B} = v_A - v_B.

1D Relative Motion:

• Same direction: v_rel = v_A - v_B (approach/receding speed)
• Opposite direction: v_rel = v_A + v_B (closing speed)

Rain-Man Problem: The velocity of rain relative to the walking man: v_{rain/man} = v_rain - v_man. The man should tilt his umbrella at angle arctan(v_man/v_rain) from vertical, in his direction of motion.

River Crossing: A swimmer with speed v_s in still water, river speed v_r, width d:

• Minimum time path: Swim perpendicular to bank. t_min = d/v_s. Drift = v_r * d/v_s.
• Zero drift path (shortest path): Angle upstream by theta = arcsin(v_r/v_s). Effective speed = sqrt(v_s^2 - v_r^2). Time = d/sqrt(v_s^2 - v_r^2). Only possible if v_s > v_r.
• If v_s < v_r: Cannot cross with zero drift. Minimum drift occurs at angle arccos(v_s/v_r) from upstream.

JEE Application: Always draw velocity triangles. Identify what is given (minimum time? zero drift? minimum drift?) and choose the appropriate case.