Velocity–Time Graphs

A story about a moving object can be drawn as a picture. On a velocity–time graph we put time along the bottom (the horizontal axis) and velocity up the side (the vertical axis). Each point on the line says: "at this moment, the object was going at that speed."

Read left to right and the line tells the whole journey — when the object sped up, when it cruised, when it slowed to a stop. But a velocity–time graph hides two treasures that aren't obvious at first glance: the steepness of the line and the space underneath it. Uncover those two and you can read off an object's acceleration and the distance it travelled — without ever watching it move.

The gradient is the acceleration

Acceleration means how quickly the velocity is changing. On the graph, that is exactly how quickly the line climbs as you move to the right — its gradient. Gradient is "how much the value goes up divided by how far you went across", so

\text{acceleration} \;=\; \text{gradient} \;=\; \frac{\text{change in velocity}}{\text{time taken}} \;=\; \frac{\Delta v}{\Delta t}.

The shape of the line therefore tells you the motion at a glance:

A line sloping up → the velocity is rising → speeding up. The steeper the slope, the harder the acceleration.
A flat, horizontal line → the velocity isn't changing → steady speed (zero gradient means zero acceleration).
A line sloping down → the velocity is falling → slowing down (a negative gradient, sometimes called deceleration).

Worked example: acceleration from the gradient

A car pulls away from a stop. Its velocity–time line rises straight from 0 to 20\ \text{m/s} in 8\ \text{s}. What is its acceleration?

Just take the gradient — the rise in velocity over the time across:

a = \frac{\Delta v}{\Delta t} = \frac{20 - 0}{8} = \frac{20}{8} = 2.5\ \text{m/s}^2.

So the car gains 2.5\ \text{m/s} of speed every second. Notice the units fall straight out of the formula: metres-per-second, divided by seconds, gives metres per second, per second — written \text{m/s}^2. A steeper line here would have meant a bigger number and a fiercer push in your seat.

The area under the line is the distance

Here is the second treasure. The area trapped between the line and the time axis is the distance travelled. Why? Because \text{distance} = \text{velocity} \times \text{time}, and on the graph velocity is the height while time is the width — multiply them and you get an area.

Steady speed makes a flat line, so the area beneath is a rectangle: \text{distance} = v \times t (height × width).
Steady acceleration from rest makes a straight slope up from zero, so the area beneath is a triangle: \text{distance} = \tfrac{1}{2} \times \text{base} \times \text{height} = \tfrac12\,b\,h.

For a whole journey — speed up, cruise, slow down — you just chop the area into rectangles and triangles, work out each, and add them up.

Worked example: distance from the area

A cyclist accelerates steadily from rest, and after 6\ \text{s} is riding at 12\ \text{m/s}. The velocity–time line is a straight slope from the origin up to that point, so the area under it is a triangle. Its base is the time (6\ \text{s}) and its height is the final velocity (12\ \text{m/s}):

\text{distance} = \tfrac{1}{2}\,b\,h = \tfrac{1}{2} \times 6 \times 12 = 36\ \text{m}.

The cyclist covered 36\ \text{m} in those six seconds. And if they then held a steady 12\ \text{m/s} for another 10\ \text{s}, that stretch is a rectangle, 12 \times 10 = 120\ \text{m} — for a running total of 36 + 120 = 156\ \text{m}.

Picture a driver's whole trip: they floor it for a few seconds (a rising line), settle into a steady motorway cruise (a long flat line), then brake for the exit (a line sloping back down to zero). Sketch that as a velocity–time graph and it makes a trapezium — slanted up one side, flat across the top, slanted down the other. Traffic engineers really do read total journey distance straight off that area, and racing teams pore over a car's velocity–time trace to see exactly where a rival braked a fraction later. The whole trip's distance is nothing more mysterious than the size of that shape.

Play with it

Below is a velocity–time graph of an object starting from rest and accelerating steadily for 8\ \text{s}. Drag the acceleration slider and watch two things at once: the line gets steeper (bigger gradient = harder acceleration), and the shaded triangle underneath grows (bigger area = greater distance). The readouts show the gradient and the area so you can check them against the formulas.

Don't mix it up with a distance–time graph

A distance–time graph and a velocity–time graph look similar but say completely different things — the axes are different, so the same line means something else on each.

On a distance–time graph the gradient is the speed, and a flat line means the object has stopped (its distance isn't changing).
On a velocity–time graph the gradient is the acceleration, the area underneath is the distance, and a flat line means a steady speed — the object is still very much moving.

The two things a velocity–time graph tells you

The gradient (steepness) of the line is the acceleration, a = \dfrac{\Delta v}{\Delta t}.
The area between the line and the time axis is the distance travelled — a rectangle for steady speed (v\,t), a triangle for steady acceleration (\tfrac12 b h).

The trap that catches almost everyone: a flat horizontal line on a velocity–time graph does not mean the object has stopped. It means the object is moving at a steady speed — the velocity isn't changing, so the acceleration is zero, but the object is still cruising along. (That flat line only means "stopped" on a distance–time graph, where staying at the same distance really is standing still.)

Two more mix-ups to avoid on a velocity–time graph: the gradient is the acceleration, not the distance, and the distance is the area, not the height of the line. Reach for the slope when you're asked "how fast is it speeding up?" and for the area when you're asked "how far did it go?"