Distance-Time and Velocity-Time Graphs

This lesson covers how to interpret and draw distance-time graphs and velocity-time graphs as required by the AQA GCSE Physics specification (4.5.6). Motion graphs are one of the most important and frequently tested topics in GCSE Physics. You must be able to read information from graphs, calculate speed and acceleration from graphs, and describe the motion shown by different sections of a graph.

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Distance-Time Graphs

A distance-time graph shows how the distance travelled by an object changes over time.

• Distance is plotted on the y-axis (vertical axis).
• Time is plotted on the x-axis (horizontal axis).

What Different Sections Mean

Graph Feature	What It Means
Horizontal line (flat section)	The object is stationary (not moving)
Straight diagonal line (going up)	The object is moving at a constant speed
Steeper line	The object is moving faster
Shallower line	The object is moving slower
Curve getting steeper	The object is accelerating (speeding up)
Curve getting shallower	The object is decelerating (slowing down)

Calculating Speed from a Distance-Time Graph

The gradient (slope) of a distance-time graph gives the speed.

Speed = gradient = change in distance / change in time

For a straight line, the gradient is constant, so the speed is constant.

For a curved line, the gradient changes. You can find the instantaneous speed at a particular time by drawing a tangent to the curve at that point and calculating the gradient of the tangent.

graph LR
    subgraph "Distance-Time Graph Features"
        A["Flat = stationary"] --> B["Diagonal = constant speed"]
        B --> C["Steeper = faster"]
        C --> D["Curve = changing speed"]
    end

    style A fill:#e74c3c,color:#fff
    style B fill:#2980b9,color:#fff
    style C fill:#27ae60,color:#fff
    style D fill:#e67e22,color:#fff

Exam Tip: To find the speed from a distance-time graph, calculate the gradient: pick two points on the straight line, and divide the change in distance by the change in time. For a curve, draw a tangent at the point of interest and find the gradient of the tangent. Always show your working with the values you read from the graph.

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Worked Example: Distance-Time Graph

A distance-time graph shows the following data:

Time (s)	Distance (m)
0	0
10	50
20	100
30	100
40	200

Section 1 (0–20 s): Straight line from (0, 0) to (20, 100).

Speed = (100 - 0) / (20 - 0) = 100 / 20 = 5 m/s (constant speed)

Section 2 (20–30 s): Horizontal line at 100 m.

Speed = 0 m/s. The object is stationary.

Section 3 (30–40 s): Straight line from (30, 100) to (40, 200).

Speed = (200 - 100) / (40 - 30) = 100 / 10 = 10 m/s (constant speed, faster than section 1)

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Velocity-Time Graphs

A velocity-time graph shows how the velocity of an object changes over time.

• Velocity is plotted on the y-axis (vertical axis).
• Time is plotted on the x-axis (horizontal axis).

What Different Sections Mean

Graph Feature	What It Means
Horizontal line	The object is moving at a constant velocity
Horizontal line at zero	The object is stationary
Straight line going up (positive gradient)	The object is accelerating at a constant rate
Straight line going down (negative gradient)	The object is decelerating at a constant rate
Steeper line	Greater acceleration or deceleration
Curve	Changing (non-uniform) acceleration

Calculating Acceleration from a Velocity-Time Graph

The gradient of a velocity-time graph gives the acceleration.

Acceleration = gradient = change in velocity / change in time

a = (v - u) / t

Where:

• a = acceleration (m/s^2)
• v = final velocity (m/s)
• u = initial velocity (m/s)
• t = time taken (s)

A positive gradient means the object is accelerating (speeding up). A negative gradient means the object is decelerating (slowing down).

Calculating Distance from a Velocity-Time Graph

The area under a velocity-time graph gives the distance travelled.

For a rectangle: area = base x height = time x velocity

For a triangle: area = 0.5 x base x height = 0.5 x time x velocity

For complex shapes, split the area into rectangles and triangles, calculate each area, and add them together.

Exam Tip: The area under a velocity-time graph equals the distance travelled. This is extremely commonly tested. For shapes that are not simple rectangles or triangles, you may need to count squares on the graph paper or use the trapezium rule. Always show your method clearly.

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Worked Example: Velocity-Time Graph

A velocity-time graph shows the following data:

Time (s)	Velocity (m/s)
0	0
5	10
10	10
15	0

Section 1 (0–5 s): Velocity increases from 0 to 10 m/s.

Acceleration = (10 - 0) / (5 - 0) = 10 / 5 = 2 m/s^2

Distance = area of triangle = 0.5 x 5 x 10 = 25 m

Section 2 (5–10 s): Velocity constant at 10 m/s.

Acceleration = 0 m/s^2 (constant velocity)

Distance = area of rectangle = 5 x 10 = 50 m

Section 3 (10–15 s): Velocity decreases from 10 to 0 m/s.

Acceleration = (0 - 10) / (15 - 10) = -10 / 5 = -2 m/s^2 (deceleration)

Distance = area of triangle = 0.5 x 5 x 10 = 25 m

Total distance = 25 + 50 + 25 = 100 m

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Comparing Distance-Time and Velocity-Time Graphs

Feature	Distance-Time Graph	Velocity-Time Graph
Gradient gives	Speed	Acceleration
Area under graph gives	Not applicable	Distance travelled
Horizontal line means	Stationary	Constant velocity
Straight line up means	Constant speed	Constant acceleration
Curve means	Changing speed	Changing acceleration

graph TD
    subgraph "Distance-Time"
        DT1["Gradient = Speed"]
    end
    subgraph "Velocity-Time"
        VT1["Gradient = Acceleration"]
        VT2["Area = Distance"]
    end

    style DT1 fill:#2980b9,color:#fff
    style VT1 fill:#27ae60,color:#fff
    style VT2 fill:#e67e22,color:#fff

Exam Tip: Many students confuse what the gradient and area represent on each type of graph. Remember: on a distance-time graph, the gradient = speed. On a velocity-time graph, the gradient = acceleration AND the area = distance. Write this at the top of your exam paper as a reminder.

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Drawing Tangents to Curves

When a graph is curved, the gradient is changing. To find the instantaneous speed (on a distance-time graph) or instantaneous acceleration (on a velocity-time graph) at a specific time:

1. Find the point on the curve at the specified time.
2. Place a ruler so it just touches the curve at that point (tangent line).
3. Draw the tangent line with a ruler.
4. Calculate the gradient of the tangent line.

The tangent should touch the curve at only one point and should not cross the curve.

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Describing Motion from Graphs

When asked to "describe the motion shown by the graph," you should:

1. Break the graph into sections (where the shape changes).
2. For each section, state whether the object is:
   • Stationary, moving at constant speed/velocity, accelerating, or decelerating.
3. Include values — calculate speeds and accelerations from the graph.
4. Include times — state the time interval for each section.

Example answer: "From 0 to 10 s, the object accelerates at 2 m/s^2 from rest to 20 m/s. From 10 to 25 s, the object travels at a constant velocity of 20 m/s. From 25 to 30 s, the object decelerates at 4 m/s^2 and comes to rest."

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Common Exam Mistakes