Position-Time Graphs

Suppose a vehicle is moving on a straight road and we record its position from a fixed origin at every second:

To turn this into a graph, follow four steps:

1. Draw two perpendicular axes. The horizontal axis is the x-axis; the vertical axis is the y-axis. Their meeting point is the origin O.
2. Decide which quantity goes on which axis. The convention for motion: time on the x-axis, position on the y-axis. (The independent variable — what you control — goes on the x-axis. You choose the times; you measure the positions.)
3. Choose a scale that uses the available space. For our data, 5 small divisions = 1 second on the x-axis and 5 small divisions = 20 m on the y-axis works well — the graph fills the page without crowding.
4. Plot each (time, position) pair as a single dot. For (1 s, 20 m), find the point that is 1 second along the x-axis and 20 m up the y-axis — and place a dot. Repeat for every row of the table.

When all dots are placed, join them with a single smooth line. That line is the position-time graph of the motion. For our data, the dots fall on a perfectly straight line — the signature of constant velocity, as we'll see in the next section.

Once a position-time graph is drawn, you can read the kind of motion straight off the shape — without any calculation. There are three shapes you must recognise instantly:

1. Straight line (going up or down) → constant velocity.

If in equal time intervals the position changes by equal amounts, the dots line up perfectly. Every second, the same step. The slope (steepness) tells you the speed; the direction (rising or falling) tells you the sign of velocity.

2. Curved line → changing velocity → the object is accelerating.

If the position changes by unequal amounts in equal time intervals — small step at first, bigger steps later — the line bends. A curve that gets steeper means the object is speeding up. A curve that flattens means the object is slowing down.

3. Horizontal straight line → object is at rest.

If the position does not change with time, the dots all sit at the same height. The line is parallel to the x-axis. The object is parked.

Memorise these three shapes. They are the alphabet of motion graphs. Almost every position-time graph you will see in this book — and in every physics exam — is some combination of these three.

A caution: a position-time graph is not a route map. It does not show the path the object travelled along the road; it shows how its position (a single number on the chosen axis) changed with time. A bus driving north for an hour and a bus driving south for an hour can have position-time graphs that are mirror images, but they are not driving on the same road.

The shape tells you what kind of motion. The slope tells you how fast.

The slope of any straight line is its steepness — how much it rises as you go across. On a position-time graph, slope is precisely:

$\text{slope} = \frac{\text{change in position}}{\text{change in time}} = \frac{s_2 - s_1}{t_2 - t_1}$

But that is exactly the formula for average velocity! So:

The slope of the position-time graph at any segment is equal to the average velocity over that segment.

This is the single most important sentence on this page.

The triangle trick. To find the slope between two points $A$ and $B$ on the line, drop a horizontal line from $A$ rightward and a vertical line from $B$ downward. You get a right triangle $\triangle ABC$ with $A$ in the top-left, $B$ in the top-right, and $C$ in the bottom-right corner.

• The horizontal leg $AC$ is the time interval $t_2 - t_1$.
• The vertical leg $BC$ is the change in position $s_2 - s_1$.
• The slope is $\frac{BC}{AC}$ — and this number, in m/s, is the velocity.

A steep line means a large slope means a high velocity. A shallow line means a small slope means a low velocity. A horizontal line has slope zero — and zero velocity, which is the same as being at rest. Three shapes, one number to compute.