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(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 = d/sqrt(v_s^2 - v_r^2)
• Shortest time: swim perpendicular to bank, time = d/v_s, drift = v_r * d/v_s