Distance–Time Graphs

Imagine walking to school. At every moment you are some distance from your front door, and the clock keeps ticking. A distance–time graph is just a way of drawing that whole journey as a single line, so you can see it all at once — where you sped up, where you dawdled, and where you stopped to tie a shoelace.

The rule for reading one never changes:

Time runs along the bottom (the horizontal axis) — usually in seconds.
Distance goes up the side (the vertical axis) — usually in metres.

So every point on the line answers one question: "how far had I travelled by this moment?" Follow the line left-to-right and you are watching time march forward and the journey unfold.

The steepness is the speed

Here is the single most important idea on the page. The gradient — how steep the line is — tells you the speed. A line that climbs quickly means lots of distance covered in a little time: that is fast. A line that barely rises means hardly any distance in all that time: that is slow.

Read a distance–time line like this:

A flat (horizontal) line → stopped. Time passes, but the distance never changes.
A gently sloping straight line → moving at a slow, steady speed.
A steep straight line → moving at a fast, steady speed.
The steeper the line, the faster the object is going.

Because the line is straight in every one of these cases, the speed is steady (constant) — the object holds one pace the whole way.

Reading the speed straight off the graph

A straight line has one gradient, and that gradient is the speed. To measure it, pick any two points on the line and work out how much the distance changed, divided by how much the time changed:

\text{speed} = \text{gradient} = \frac{\text{change in distance}}{\text{change in time}} = \frac{\Delta d}{\Delta t}

(The little triangle \Delta just means "the change in".) This is exactly the "rise over run" you use for the gradient of a line in maths — here the rise is distance and the run is time.

Worked example. A runner's line climbs steadily from 0\,\text{m} to 12\,\text{m} while the clock goes from 0\,\text{s} to 4\,\text{s}. The distance changed by \Delta d = 12\,\text{m} and the time by \Delta t = 4\,\text{s}, so

\text{speed} = \frac{\Delta d}{\Delta t} = \frac{12\,\text{m}}{4\,\text{s}} = 3\,\text{m/s}.

The units come along for free: metres divided by seconds gives metres per second (m/s). Steeper line, bigger fraction, bigger speed.

Play with the gradient

Below is a live distance–time graph. Drag the speed slider and watch the line tilt: crank the speed up and the line rears up steeply; drop it to zero and the line lies flat — the object is stopped. Drag the time slider to send the dot travelling along its journey; the dashed triangle shows the run (time so far) along the bottom and the rise (distance so far) up the side, whose ratio is the speed printed at the top.

When the line bends: changing speed

A straight line meant one steady speed. But a real journey speeds up and slows down, and then the line curves. The trick is to keep watching the steepness, moment by moment:

A curve that gets steeper and steeper → the object is speeding up (each second it covers more ground than the last).
A curve that flattens off (gets less steep) → the object is slowing down.
Where the curve momentarily goes flat → the object has, for that instant, stopped.

Think of a car pulling away from traffic lights: it starts flat-ish (slow), then the line curls upward more and more sharply as it accelerates onto the open road. You read a curve the same way you read a straight line — by its steepness — you just have to do it all along its length.

Two traps snare almost everyone the first time:

A flat line does NOT mean "going backwards". A horizontal line means the distance is not changing — the object is stationary, sitting still while time ticks on. It is standing where it is, not reversing.
A line sloping back DOWN to zero means "returning to the start". When the distance decreases, the object is heading back towards where it began; reaching 0\,\text{m} again means it is home. (A steeper downward slope just means it is hurrying back faster.)

And remember: a distance–time graph is not a picture of the route. A perfectly straight, rising line does not mean the road was straight — it means the speed was steady.

Sam cycles to the shop. For the first 20\,\text{s} the line climbs steeply and steadily to 100\,\text{m} — a brisk, constant speed of \tfrac{100}{20} = 5\,\text{m/s}. Then the line goes dead flat for 30\,\text{s}: that is Sam parking the bike and buying a drink — stopped, distance frozen at 100\,\text{m}. Finally the line slopes gently back down to 0\,\text{m}: Sam is freewheeling home, slower than before because the slope is gentler.

Notice you never saw a single photograph of Sam — yet from three line segments you can retell the entire outing, complete with the shop stop. That is the quiet power of a distance–time graph.