Relative Velocity
Overview
The velocity of A relative to B is: v_A/B = v_A − v_B. This is a vector subtraction. Example: if a train moves at 30 m/s east and a car moves at 20 m/s east, the train's velocity relative to the car = 30 − 20 = 10 m/s east. If they move in opposite directions, the relative speed is the sum of the magnitudes.
What does 'relative velocity' mean?
The velocity of object A relative to object B is what the velocity of A looks like to an observer sitting on B. Mathematically: v_A/B = v_A − v_B (vector subtraction). If you are in a train moving at 30 m/s east and a car next to the track moves at 20 m/s east, the train's velocity relative to the car is 30 − 20 = 10 m/s east. From the car's perspective, the train appears to move at only 10 m/s.
Opposing motion and river crossing
When two objects move in opposite directions, their relative speed is the sum of their speeds. For a river crossing: if a boat aims straight across a river of width d flowing at speed v_r, and the boat's speed relative to water is v_b, the actual velocity of the boat relative to the ground is the vector sum of v_b and v_r. To cross in minimum time, point straight across. To cross with zero drift, angle the boat upstream.
- ⚠Forgetting the subtraction is a vector subtraction — direction matters
- ⚠Mixing up 'velocity of A relative to B' with 'velocity of B relative to A' — they are opposite in direction