Relative velocity in 1D

Intermediate

Velocity of A as seen from B is the difference of velocities.

Key relationv_AB = v_A − v_B

The velocity of A relative to B is v_AB = v_A − v_B in the same direction convention. Overtaking and head-on motions become simple by working in the frame of one body.

Worked example

Cars at 30 m/s and 20 m/s in the same direction: relative speed 10 m/s. Head-on: 50 m/s.

Common mistake

Define which direction is positive before you subtract.

In more detail

Relative velocity is the velocity of one object as seen from another. In one dimension, vAB = vA − vB.

How to solve it

Choose a positive direction and label each velocity with its sign.
Subtract the velocity of the reference object from the velocity of the object you are interested in: v_AB = v_A − v_B.
Use the relative velocity as if the reference object were at rest.
Divide the separation by the relative speed to find the time until they meet or pass.

Worked examples

Example 1: Train A moves at 25 m/s and train B at 15 m/s, both in the same direction. A is 200 m behind B. How long until A catches up?

v_AB = 25 − 15 = 10 m/s. Time = 200/10 = 20 s.

Example 2: A car travels east at 20 m/s and a second travels west at 30 m/s. They start 1 km apart and drive towards each other. When do they meet?

Use east as positive: v_car1 = +20, v_car2 = −30. Relative velocity = 20 − (−30) = 50 m/s. Time = 1000/50 = 20 s.

Checking your answer

When the two objects move in the same direction, the relative speed is smaller than either speed. When they move towards each other, it is larger. Both examples follow this pattern.

Real-world uses

Two cars travelling in the same direction at different speeds approach each other at the difference of their speeds.
Overtaking manoeuvres are judged using relative velocity.
When two trains pass in opposite directions, the speed of one as seen from the other is the sum of their speeds.

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Frequently asked questions

Is relative velocity always subtraction?

In one dimension along a line, yes. In two dimensions you must subtract vectors.

Does relative velocity depend on the frame?

The result depends on the frame used to describe both velocities, but the rule is the same.

What is the relative velocity of two objects moving the same way?

Subtract the velocities. Two cars at 20 m/s and 15 m/s move apart at 5 m/s relative to each other.