# Motion Graphs

> Reading distance-time, speed-time and acceleration-time graphs. Gradients and areas tell the whole story.

From HSC Physics on WhizzNotes, free study notes for NSW students: https://whizznotes.com/physics/motion-graphs/. Updated 2 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The dog walk
Every afternoon Kai takes his dog, Biscuit, to the park. Kai's phone app tracks the walk and draws a graph afterwards.

The line goes up steadily as they walk to the park. Then it goes flat for a while. That was when Biscuit stopped to sniff a tree for five whole minutes. Then the line shoots up steeply. A cat had appeared, and Biscuit dragged Kai along at a sprint.

Kai never wrote a single word, yet the graph tells the whole story. Once you can read motion graphs, you can read stories like this too.

**In short:** Motion graphs show how distance, speed or acceleration change with time. On a distance-time graph the gradient is the speed. On a speed-time graph the gradient is the acceleration and the area under the line is the distance travelled.

## Distance-time graphs

Time goes on the x-axis. Distance goes on the y-axis.

*Diagram: Three small distance-time graphs: a flat line meaning stopped, a straight sloping line meaning constant speed, and a curve getting steeper meaning speeding up*
*Distance-time graphs: flat, straight slope, curve getting steeper.*

| What you see | What it means |
|---|---|
| Flat (horizontal) line | Not moving |
| Straight sloping line | Constant speed |
| Steeper line | Faster |
| Curve getting steeper | Speeding up (accelerating) |
| Curve getting flatter | Slowing down |

### Key idea: Gradient of a distance-time graph = speed
$$\text{speed} = \text{gradient} = \frac{\text{rise}}{\text{run}} = \frac{\Delta d}{\Delta t}$$

### Worked example: Reading speed from a graph
A straight line on a distance-time graph goes from (0 s, 0 m) to (20 s, 100 m).

Speed $= \dfrac{100 - 0}{20 - 0} = 5 \text{ m/s}$.

## Speed-time (velocity-time) graphs

Now **speed** goes on the y-axis. This changes everything, so always check the y-axis label first.

*Diagram: Three small speed-time graphs: a flat line meaning constant speed, a rising straight line meaning constant acceleration, and a falling line meaning slowing down*
*Speed-time graphs: a flat line here means steady speed, not stopped.*

### Key idea: Two facts to remember
- **Gradient** of a speed-time graph = **acceleration**: $a = \dfrac{\Delta v}{\Delta t}$
- **Area under** a speed-time graph = **distance travelled**

### Worked example: Area for distance
A car speeds up steadily from 0 to 20 m/s in 10 s, then stays at 20 m/s for 5 s.

- Acceleration (first part): $\dfrac{20}{10} = 2 \text{ m/s}^2$
- Distance in first part (triangle): $\tfrac{1}{2} \times 10 \times 20 = 100$ m
- Distance in second part (rectangle): $5 \times 20 = 100$ m
- Total distance: **200 m**

### Watch out: Flat does not always mean stopped
A flat line on a **distance-time** graph means stopped. A flat line on a **speed-time** graph means moving at a steady speed. Check the axis every time.

## Acceleration-time graphs

Acceleration goes on the y-axis.

- A flat line **above** zero means constant acceleration.
- A flat line **on** zero means constant velocity (no acceleration).
- A flat line **below** zero means constant deceleration.
- **Area under** an acceleration-time graph = **change in velocity**.

### Worked example: A falling skydiver
Just after jumping, a skydiver's acceleration is about 9.8 m/s². As she speeds up, air resistance grows, so the acceleration gets smaller. Eventually air resistance equals her weight and acceleration drops to zero. She now falls at a steady speed called **terminal velocity**. On an a-t graph this is a curve falling from 9.8 down to 0.

**Mind map**

```text
Motion graphs
  Distance-time
    Gradient = speed
    Flat = stopped
    Curve = changing speed
  Speed-time
    Gradient = acceleration
    Area = distance
    Flat = constant speed
  Acceleration-time
    Area = change in velocity
    Flat on zero = constant velocity
  Tips
    Read the y-axis first
    Units on axes
```

### Year 11: Displacement and velocity graphs with direction
In Year 11 the y-axes become **displacement** and **velocity**, which can be **negative**.

- On a **velocity-time** graph, a line below the x-axis means moving backwards (in the negative direction).
- The area below the axis counts as **negative displacement**. Add areas with signs to find the total displacement. Add their sizes (all positive) to find total distance.
- For a curved displacement-time graph, the **instantaneous velocity** at a point is the gradient of the **tangent** to the curve at that point.

A ball thrown straight up has a velocity-time graph that is a straight line sloping down, crossing zero at the top of its flight. The gradient is $-9.8 \text{ m/s}^2$ the whole time, even at the top.

### Year 12: Graphs as calculus
The links between the graphs are calculus:

$$v = \frac{ds}{dt} \qquad a = \frac{dv}{dt} \qquad s = \int v\,dt \qquad \Delta v = \int a\,dt$$

"Gradient" is differentiation. "Area under" is integration. If $s = 3t^2$, then $v = 6t$ and $a = 6 \text{ m/s}^2$, a constant acceleration.

## Quick summary

- **Distance-time**: gradient = speed. Flat = stopped. Steeper = faster.
- **Speed-time**: gradient = acceleration. Area under = distance. Flat = steady speed.
- **Acceleration-time**: area under = change in velocity.
- Always read the y-axis label before you read the shape.
- Area of a triangle = $\tfrac{1}{2} \times \text{base} \times \text{height}$. Area of a rectangle = base × height.

## Quiz

Try these questions from the chapter quiz.

1. On a distance-time graph, a horizontal line means the object is:
   - A. Stationary
   - B. Moving at constant speed
   - C. Accelerating
   - D. Decelerating
   - Answer: A. Stationary. Distance is not changing, so the object is not moving.

2. On a speed-time graph, a horizontal line means the object is:
   - A. Stationary
   - B. Moving at constant speed
   - C. Accelerating
   - D. Decelerating
   - Answer: B. Moving at constant speed. Speed is not changing, so the object moves steadily.

3. The gradient of a distance-time graph gives:
   - A. Acceleration
   - B. Speed
   - C. Distance
   - D. Time
   - Answer: B. Speed. Rise over run is distance ÷ time, which is speed.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/physics/motion-graphs/#quiz