Relative Motion and Relative Velocity

Relative velocity is the velocity of one object as measured from another moving object, found by vector subtraction: the velocity of A relative to B is v_AB = v_A − v_B, where both velocities are measured in the same reference frame. In brief: in the same direction, subtract the speeds; in opposite directions, add them; at right angles, use the Pythagorean theorem. For a boat in a river or a plane in wind, the velocity relative to the ground is the vector sum of the velocity relative to the medium and the velocity of the medium.

When a faster car overtakes a truck, the truck appears to move slowly to the driver. A swimmer crossing a river is carried downstream. Both are problems of relative motion. This guide explains relative velocity in one and two dimensions, compares the cases in a table, works through examples with cars and a river crossing and corrects common mistakes.

What is relative velocity?

All velocities are measured relative to a frame of reference. The velocity of a passenger walking in a train relative to the ground is the velocity of the train relative to the ground plus the velocity of the passenger relative to the train. In general:

v_A/ground = v_A/B + v_B/ground, so v_A/B = v_A/ground − v_B/ground.

These relations hold at speeds that are small compared with the speed of light. At speeds close to the speed of light, special relativity changes the rule for adding velocities.

How does it work in one dimension?

• Same direction: a car at 25 m/s behind a truck at 20 m/s approaches it at 25 − 20 = 5 m/s relative to the truck.
• Opposite directions: two cars moving toward each other at 25 m/s and 20 m/s approach each other at 25 + 20 = 45 m/s.
• With signs: choose one direction as positive and calculate v_A − v_B with the signs. If A moves at +25 m/s and B at −20 m/s, then v_A/B = 25 − (−20) = 45 m/s.

How does it work in two dimensions?

Add or subtract the velocities as vectors, using the head-to-tail method or components. A boat that heads straight across a river has a velocity relative to the water, u, perpendicular to the bank, and the river current has velocity w along the bank. The velocity relative to the ground is the vector sum, with magnitude √(u² + w²). The crossing time depends only on the component of velocity perpendicular to the bank (the width divided by u), while the drift downstream equals w times the crossing time. To reach a point directly opposite, the boat must be aimed upstream at an angle θ, where sin θ = w ÷ u (possible only if u > w). Planes in wind work in the same way, with the air as the moving medium.

Cases compared

Worked example: overtaking and crossing a river

1. Overtaking. A car at 25 m/s is 60 m behind a truck going at 20 m/s in the same direction.
   • Relative speed of the car to the truck: 25 − 20 = 5 m/s.
   • Time to close the 60 m gap: 60 ÷ 5 = 12 s.
   • Check: in 12 s the car travels 300 m and the truck 240 m, a difference of 60 m.
2. River crossing. A river is 100 m wide, with a current of 3.0 m/s. A boat heads directly across, with a speed of 4.0 m/s relative to the water.
   • Speed relative to the ground: √(4.0² + 3.0²) = √25 = 5.0 m/s.
   • Crossing time: 100 ÷ 4.0 = 25 s.
   • Drift downstream: 3.0 × 25 = 75 m.
   • Angle of the path to the bank direction across: tan θ = 3.0 ÷ 4.0, so θ = 37° downstream from the straight line across.
3. To land directly opposite: sin θ = 3.0 ÷ 4.0 = 0.75, so θ = 49° upstream from the perpendicular. The speed across the river is 4.0 × cos 49° = 2.6 m/s (equivalently √(16 − 9) = 2.65 m/s), so the crossing takes 100 ÷ 2.65 = 38 s, which is longer than when heading straight across.

How did the idea of relativity of motion develop?

Galileo described in 1632 how events inside the cabin of a ship look the same whether the ship is moving uniformly or not. Newtonian mechanics uses the Galilean transformation between frames. Einstein showed in 1905 that for speeds approaching that of light the rule v + w for adding velocities must be replaced by a relativistic formula, so that no speed exceeds c.

Which mistakes are common?

Mistake 1: Adding speeds when the velocities are at an angle

Use vector addition. Perpendicular velocities are combined by the Pythagorean theorem.

Mistake 2: Forgetting the frame of reference

State which object the velocity is measured against. A velocity relative to the water differs from that relative to the shore.

Mistake 3: Thinking that the shortest crossing time gives the shortest path

Heading straight across takes the least time but causes the largest drift. The shortest path requires aiming upstream.

Mistake 4: Using the Galilean rule for speeds close to light

At such speeds relativistic velocity addition must be used.