Definition: Velocity of A with respect to B: $v_{A/B}$ = $v_A$ - $v_B$ (vector subtraction)

1D Case:

• Same direction: $v_{rel}$ = $v_A$ - $v_B$ (closing speed if approaching)
• Opposite direction: $v_{rel}$ = $v_A$ + $v_B$ (closing speed)

2D Rain-Man Problem: Rain falls vertically with $v_R$. Man walks with $v_M$ horizontally.

• Velocity of rain w.r.t. man: $v_{R/M}$ = $v_R$ - $v_M$
• The man sees rain coming at angle theta = arctan$\frac{v_M}{v_R}$ from vertical
• He must tilt his umbrella at this angle in the direction of his motion

River Crossing:

• Shortest path (directly across): swim at angle upstream, time = $\frac{d}{sqrt}$($v_s^2$ - $v_r^2$)
• Shortest time: swim perpendicular to bank, time = $\frac{d}{v_s}$, drift = $v_r$ * d/$v_s$