Reference frame, trajectory and distance travelled

A bus passenger says, ‘I'm staying in the same place,’ while someone at the bus stop sees the passenger moving away. Who is wrong? Neither is, provided each says what they are measuring the motion relative to.

Place a ball on the seat of a moving bus. The distance between the ball and the seat stays the same, while houses pass the window. Mechanical motion is a change in an object's position relative to a chosen reference object as time passes. Relative to the seat, the ball is at rest; relative to the road, it is moving. Changing the description has not changed the ball. We have changed the object against which we compare its position.

How can we say where a bus is on a long, straight road? Choose a visible pole as the origin and one direction along the road as positive. The coordinate x = 20 м means a position 20 m from the origin in that direction; on the other side, the coordinate is negative. A number alone, without an origin and a direction, does not specify a position. Moving the origin changes the coordinates, but the distance between two stationary objects stays the same.

Suppose we know that the bus is beside the pole. Did it pass quickly, or stop there for a minute? A coordinate alone cannot tell us. We must record position together with time, and agree when we start measuring time. A reference object, a coordinate system fixed to it, and a clock form a reference frame. This turns ‘it's going somewhere’ into a record we can check: where the object was at each chosen instant.

Imagine marking the ball's position at short, equal time intervals and joining the dots. The line along which it moves is its trajectory. To describe the motion of a whole object, we often follow just one point, such as the centre of the ball. Drop a ball inside a bus travelling steadily: a passenger sees a nearly vertical trajectory, while someone beside the road sees a curve. Like position, the trajectory depends on the reference frame.

Lay a thread along the entire route travelled, then measure its length. This gives the distance travelled, s, measured in metres. Add the lengths of every section: turning back does not make the total decrease. Walk from the door to a tree and back. Your coordinate returns to its original value, but your feet have covered both sections. ‘Where did you end up?’ and ‘How far did you travel?’ therefore call for different quantities. On a graph of cumulative distance, the return journey adds distance too.

A student walks 30 m in a straight line from a door, then returns to the door. Find the distance travelled and the final coordinate, taking the door as the origin.

The outward section is 30 m long. The return section is also 30 m long.

The total distance is 30 + 30 = 60 м. The final position is at the origin.

Distance travelled: 60 m; final coordinate: 0 m. Returning to the door restores the original position, but does not undo either section of the walk.

Motion and rest are described relative to a chosen reference frame. Changing the observer can even change the shape of the trajectory.

Coordinates tell us where an object is; a clock lets us describe how its position changes.

Distance travelled adds the lengths of all sections. Returning to the start does not undo the distance already covered.

Distance and time let us compare how fast objects move. The next lesson introduces speed and graphs.