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Relative VelocityJEE Main

Velocity of A as seen from B — interactive Physics simulation for IIT-JEE.

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Concept

The velocity of A relative to B is a plain vector subtraction: v⃗AB=v⃗A−v⃗B\vec v_{AB} = \vec v_A - \vec v_B. It answers what B sees. Every rain–man, river–boat, and chase problem is this one operation dressed in a story.

Key formula

v⃗AB=v⃗A−v⃗B,∣v⃗AB∣=vA2+vB2−2vAvBcos⁡θ\vec{v}_{AB} = \vec{v}_A - \vec{v}_B, \qquad |\vec v_{AB}| = \sqrt{v_A^2 + v_B^2 - 2v_Av_B\cos\theta}

Derivation

Positions subtract: r⃗AB=r⃗A−r⃗B\vec r_{AB} = \vec r_A - \vec r_B. Differentiate once and the same relation holds for velocities (and again for accelerations).

Rain–man: rain falls at v⃗R\vec v_R (down), man walks at v⃗M\vec v_M (horizontal). He must tilt his umbrella along v⃗RM=v⃗R−v⃗M\vec v_{RM} = \vec v_R - \vec v_M, i.e. at tan⁡α=vM/vR\tan\alpha = v_M/v_R from the vertical, into his own motion.

Scenarios to explore

  • Relative Velocity — v(A rel B) = v_A − v_B — rain-man & river-boat in one picture.

Real-world applications

  • Crossing a river in shortest time (aim straight) vs shortest path (aim upstream).
  • Aircraft wind-triangle navigation.
  • Closing-speed analysis in collision avoidance.

JEE exam tips

  • Shortest river-crossing time: aim perpendicular; drift = (u/v)·d downstream.
  • To cross by the shortest PATH, aim upstream with sinθ = u/v — possible only if v > u.
  • Two objects meet when their relative velocity points along the line joining them.

Common mistakes

  • Adding the vectors instead of subtracting.
  • Tilting the umbrella away from the walk direction instead of into it.
  • Mixing frames mid-solution — pick ground or B's frame and stay there.

Exam traps to avoid

  • Minimum separation problems: minimise |r_AB|² using calculus in the relative frame, not the ground frame.
  • Rain appears vertical to a runner ⇒ horizontal rain component equals the runner's speed.