Position, Displacement and the Point-Object Model

What describing motion means

Imagine watching a bus travel along a straight road. To describe its motion, you need to say where it is and when it is there. Kinematics describes these changes without yet asking which forces cause them. Motion means a change of position with time relative to a chosen reference frame. A passenger can be at rest relative to the bus while moving relative to the road. Neither description is a contradiction: the reference frame has changed.

Rectilinear motion is motion along a straight line. Once the line is chosen, a single coordinate x specifies position. Use an origin O and declare one direction positive. If east is positive, a shop 30 m east of O has x = +30 m and a house 20 m west has x = −20 m. Negative position does not mean negative distance; it identifies a side of the origin. A clock supplies the time coordinate. Its zero can be the start of an observation, not necessarily the start of all motion.

When the size of the object can be ignored

A point object is a model in which the body is represented by one location. This works when its dimensions are small compared with the distances relevant to the question and when its orientation, rotation or shape does not affect the requested description. A railway carriage travelling between distant stations can often be represented by a point. If the question is when the whole carriage clears a short platform, its length matters and cannot be discarded.

A monkey riding on a smoothly moving cyclist may also be treated as a point when describing the overall journey on a much larger track. A curved track does not prevent the point-object approximation, although the motion is not rectilinear. A spinning ball that turns sharply at a bounce and a tumbling beaker cannot be modelled as points if the question concerns the spin, change of orientation or detailed contact. Their centre-of-mass motion can still be described separately. Always judge the approximation against the question, not just the object’s name.

Displacement compares endpoints

Suppose a walker moves from x = −20 m to x = +30 m. The displacement is +50 m: the final position lies 50 m in the positive direction from the starting position. Reversing the journey gives −50 m. Changing the origin adds the same constant to both positions, so it does not change displacement. Reversing the positive direction reverses the sign of displacement, while leaving the physical journey unchanged.

Distance adds the entire route

Distance travelled is total path length. Divide a journey at every reversal and add the non-negative lengths of all segments. For one-dimensional motion, a segment without a reversal has length equal to the magnitude of its displacement. Distance is non-negative and depends on the path; displacement may be positive, negative or zero and depends only on endpoints. Do not call a negative coordinate a negative distance travelled.

A position-time graph is a record, not a map

Put time on the horizontal axis and position on the vertical axis. Each plotted point answers “where was the object at this time?” An upward segment means position increases; a downward segment means position decreases. A horizontal segment means position stays fixed, so the object is at rest. The slope measures velocity, which the next lessons develop. A curved position-time graph does not mean the road is curved: the vertical axis is position, not a second spatial direction.

The graph shows three parts of one straight-line journey. At 3 s and 5 s, the object is at the same position because it rests between those times. At the end, its displacement is 2 m but its total distance is 10 m. A graph can contain many different times at the same position; a single object cannot have two different positions at one time.