Motion is relative — a velocity only means anything with respect to a chosen frame. The velocity of A as measured by B is v_AB = v_A − v_B (a vector subtraction), and it satisfies v_AB = −v_BA. The single trick that unlocks this whole topic is: sit in one body's frame (subtract its velocity from everything), which makes that body stationary and turns a two-body problem into a one-body problem.
1 — One dimension. Along a line, signs do the work: two bodies moving the same way have relative speed |v_A − v_B|; moving oppositely, v_A + v_B. A train/car crossing another (or a platform/pole) takes time = (sum of the lengths)/(relative speed) — same direction subtracts speeds, opposite adds.
2 — Two dimensions. v_AB = v_A − v_B as vectors: magnitude √(v_A² + v_B² − 2·v_A·v_B·cosθ). Two perpendicular motions give √(v_A² + v_B²).
3 — The river-boat problem. A boat of speed v_b in a stream of speed v_c crossing a river of width w:
- Minimum time: point the boat straight across. Then
t = w/v_b (the current doesn't affect the crossing time), but the current sweeps it a drift = v_c · t downstream; the ground speed is √(v_b² + v_c²).
- Shortest path (zero drift, straight across the ground): aim upstream at angle
θ with sinθ = v_c/v_b; the effective across-speed is √(v_b² − v_c²) and t = w/√(v_b² − v_c²) — only possible if v_b > v_c.
- If
v_b < v_c you cannot reach the point straight across; the minimum drift is obtained by aiming upstream at cosθ = v_b/v_c.
4 — The rain-man problem. To keep dry, hold the umbrella along the rain's velocity relative to you, v_rain − v_man. Rain falling vertically at v_r while you walk at v_m appears to come at tanθ = v_m/v_r from the vertical, with apparent speed √(v_r² + v_m²). Aeroplane-in-wind is the same geometry: head into the wind to cancel its cross-component.
5 — Approach, meeting and closest distance. Time to meet = (initial separation)/(relative speed of approach). For the closest distance of approach of two particles, go to the relative frame: one particle sits still while the other travels in a straight line with velocity v_rel; the minimum separation is the perpendicular distance from the stationary particle to that line, d·sinφ, reached at time (d·cosφ)/|v_rel|. (If that perpendicular distance is zero, they collide.) …