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: $v_{rel}$ = $v_A$ - $v_B$ $\frac{approach}{receding speed}$
• 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$\frac{v_man}{v_rain}$ 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. $t_{min}$ = $\frac{d}{v_s}$. Drift = $v_r$ * d/$v_s$.
• Zero drift path (shortest path): Angle upstream by theta = arcsin$\frac{v_r}{v_s}$. Effective speed = sqrt($v_s^2$ - $v_r^2$). Time = $\frac{d}{sqrt}$($v_s^2$ - $v_r^2$). Only possible if $v_s$ > $v_r$.
• If $v_s$ < $v_r$: Cannot cross with zero drift. Minimum drift occurs at angle arccos$\frac{v_s}{v_r}$ from upstream.

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