Speed-Time Graphs

A different kind of motion graph

You may already know how to read a distance-time graph, where the line shows how far an object has gone. A speed-time graph answers a different question: not how far, but how fast — and how that speed changes from one moment to the next.

This small change of axis makes the graph extremely powerful. From it you can read two completely different quantities: the acceleration (from the slope) and the distance (from the area underneath). Learning to extract both is a key skill in physics.

Setting up the axes

A speed-time graph is built like this:

• Time goes along the bottom — the x-axis — in seconds.
• Speed goes up the side — the y-axis — in metres per second (m/s).

The height of the line tells you how fast the object is going at that moment. A line high up means a fast speed; a line near the bottom means a slow speed.

Reading the shape

Each shape on a speed-time graph has a clear meaning:

• A flat (horizontal) line → the speed is not changing → constant speed (the object is still moving).
• A line on the time axis (speed = 0) → the object is stationary.
• An upward slope → the speed is increasing → the object is accelerating.
• A downward slope → the speed is decreasing → the object is decelerating.
• A steeper slope → a greater acceleration (speed changing more quickly).

Notice the big difference from a distance-time graph: here a flat line means constant speed, not stopped.

Gradient = acceleration

The slope of a speed-time graph tells you the acceleration — how quickly the speed is changing. It is worked out as:

acceleration = change in speed ÷ change in time

The units are metres per second per second, written m/s². An acceleration of 2 m/s² means the speed increases by 2 m/s during every second. If you would like a fuller explanation of this idea, see acceleration explained.

Area under the line = distance

Here is the second power of the graph. The area between the line and the time axis equals the distance travelled. This works because distance = speed × time, and area = height × width on the graph.

• For a flat line, the area is a simple rectangle: area = speed × time.
• For a sloping line starting from rest, the area is a triangle: area = ½ × base × height.
• For a more complex shape, split it into rectangles and triangles and add the areas together.

Worked examples

Example 1 — acceleration. A train speeds up from 0 m/s to 30 m/s in 15 seconds. The graph is a straight line rising from the origin. What is its acceleration?

acceleration = change in speed ÷ time = (30 − 0) ÷ 15 = 2 m/s²

Example 2 — distance from a triangle. Using the same train, what distance did it cover during those 15 seconds?

The shape under the line is a triangle. area = ½ × base × height = ½ × 15 × 30 = 225 m

Example 3 — a three-part journey. A car accelerates from 0 to 12 m/s in 4 s, holds 12 m/s for 6 s, then brakes to 0 in 3 s. Find the total distance.

Part 1 (triangle): ½ × 4 × 12 = 24 m Part 2 (rectangle): 12 × 6 = 72 m Part 3 (triangle): ½ × 3 × 12 = 18 m Total distance = 24 + 72 + 18 = 114 m

Splitting the area into simple shapes turns a tricky problem into easy arithmetic.

Why this matters

Speed-time graphs are used everywhere engineers care about how things speed up and slow down — designing brakes, planning how fast a lift can safely accelerate, or working out the runway length a plane needs for take-off. Being able to pull both acceleration and distance from one graph is exactly why physicists love them.

Try it yourself! 🧪

Map the speed of a toy car rolling down a ramp.

You need a smooth ramp (a board on some books), a small toy car, a tape measure, and a phone that can record slow-motion video.

1. Mark the ramp every 10 cm from the top.
2. Record a slow-motion video of the car rolling down from rest.
3. Play it back frame by frame. Note the time the car reaches each mark.
4. Work out the speed over each 10 cm section (distance ÷ time for that section).
5. Plot speed up the side and time along the bottom.

You should see an upward-sloping line — the car is accelerating as gravity pulls it down the ramp. Estimate the gradient to find its acceleration, and estimate the area under the line to find the distance it rolled. You have just measured acceleration with nothing but a toy car and a phone!