Trajectory relative to the reference frame

The trajectory changes too!

The trajectory is the set of successive positions occupied by an object during its motion. Like the motion itself, the trajectory depends on the chosen reference frame.

A classic example: a bicycle tyre valve

Let’s take the valve of a bicycle wheel rolling straight along a road.

• In the cyclist’s frame of reference (attached to the bicycle): the valve rotates in a circle around the wheel’s axis. Its trajectory is a circle.
• In the Earth’s reference frame (the ground, as seen by a stationary pedestrian): the valve moves forwards whilst rotating. Its trajectory resembles a series of loops, known as a cycloid.

It is the same object, the same actual motion, but two completely different trajectories depending on where you observe it from!

Summary table

Situation	Reference frame	Observed trajectory
Fall of a ball dropped from a moving train	Train’s reference frame	Vertical straight line
Same fall	Earth’s reference frame (station platform)	Curved line (parabola)
Bicycle wheel valve	Bicycle’s reference frame	Circle
Bicycle wheel valve	Earth’s reference frame	Cycloid (loops)

Key points

A trajectory is not a fixed property of an object: it depends entirely on the frame of reference chosen to observe it. A simple trajectory (straight line, circle) in one frame of reference may become complex in another.