Relative Motion

You’ll be able to

Express velocities relative to different reference frames and convert between them
Apply the relative-velocity addition rule in one and two dimensions
Solve river-crossing and crosswind problems using vector components

Velocity depends on who is watching

There is no single "true" velocity — every velocity is measured relative to some reference frame. A passenger walking up a moving train has one velocity relative to the train and a different one relative to the ground. To translate between frames you add the velocity of one frame relative to the other.

Relative-velocity addition

v(A/C) = v(A/B) + v(B/C)

Read the subscripts like a chain: A-relative-to-C equals A-relative-to-B plus B-relative-to-C. The inner label B cancels. Reversing a pair flips the sign: v(B/A) = −v(A/B).

In two dimensions, add the vectors

The rule is a vector equation, so in 2D you add components. A boat aimed straight across a river still drifts downstream because its velocity relative to the ground is the vector sum of its velocity relative to the water and the water’s velocity relative to the ground. The across-stream and along-stream motions are independent, exactly like a projectile’s components.

Worked example

A boat can move at 4 m/s relative to the water and is aimed straight across a 100 m wide river. The current flows downstream at 3 m/s relative to the ground. Find the boat’s speed relative to the ground, the time to cross, and how far downstream it drifts.

1.Set components: across-stream velocity = 4 m/s (boat relative to water), downstream velocity = 3 m/s (current). These are perpendicular.
2.Ground speed is the magnitude: √(4² + 3²) = √(16 + 9) = √25 = 5 m/s.
3.Only the across-stream component (4 m/s) shortens the 100 m width, so crossing time = 100 / 4 = 25 s.
4.Downstream drift = current speed × time = 3 × 25 = 75 m.

Answer: Ground speed 5 m/s, crossing time 25 s, downstream drift 75 m

Checkpoint

Car A moves at +30 m/s and car B moves at −20 m/s along the same straight road (they approach each other head-on). What is the velocity of A as measured by an observer in car B?

Checkpoint

A boat heads straight across a river at 6 m/s relative to the water while the current carries it downstream at 8 m/s relative to the ground. What is the boat’s speed relative to the ground?

On the exam

Keep the subscript bookkeeping strict: write every velocity as v(object/frame). The chain v(A/C) = v(A/B) + v(B/C) only works when the inner labels match, and swapping any pair introduces a sign flip. Getting the subscripts right turns most relative-motion questions into simple addition.

Answer the 2 checkpoints as you read.