9.4 Analysing Motion Graphs

A free GCSE Physics lesson for AQA (8463) from OBA School: gradients and areas of distance–time and velocity–time graphs, tangents for changing speed, and describing whole journeys. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with every video, practice quizzes and printable worksheets with mark schemes, in our free online course.

What you will learn

• Find the speed of an object from the gradient of a distance–time graph
• (Higher tier) find the speed at an instant by drawing a tangent to a curved distance–time graph
• Find acceleration from the gradient of a velocity–time graph
• (Higher tier) find distance travelled from the area under a velocity–time graph and calculate average speed
• Use v² − u² = 2as, and describe a journey from either type of graph

Key words

• Gradient triangle: A right-angled triangle drawn on a straight line: gradient = height ÷ base.
• Tangent: (Higher tier) A straight line that touches a curve at one point without crossing it there; it has the same gradient as the curve at that point.
• Instantaneous speed: The speed at one moment in time, found from the gradient of the tangent.
• Uniform acceleration: Constant acceleration: the velocity–time graph is a straight sloping line.
• Average speed: Total distance travelled ÷ total time taken.

Speed from a distance–time graph

When the speed is constant the distance–time graph is a straight line. Draw a large right-angled triangle with the line as its slope. The height gives the distance travelled, the base gives the time taken:

speed = gradient = height of triangle ÷ base of triangle = change in distance ÷ change in time

Worked example: speed of the cyclist

gradient = 180 m ÷ 30 s

Answer: 6.0 m/s

Speed at an instant: tangents (Higher tier)

If the speed changes, the distance–time graph is curved. The gradient is different at each point, so to find the speed at a particular time:

• Mark the point on the curve at that time.
• With a ruler, draw a tangent: a straight line that just touches the curve at that point, with the curve on one side of it.
• Make the tangent long, draw a large triangle on it, and calculate its gradient. This is the speed at that instant.

Worked example: speed of the cable car at 6 s

The tangent passes through (3 s, 0 m) and (11 s, 48 m).
gradient = (48 − 0) ÷ (11 − 3) = 48 ÷ 8

Answer: 6.0 m/s at t = 6 s

Exam tip: A curve getting steeper means the speed is increasing (accelerating); a curve getting flatter means the speed is decreasing.

Acceleration and distance from a velocity–time graph

The gradient of a velocity–time graph gives the acceleration. (Higher tier) The area under it gives the distance travelled.

Worked example: the car joining the road

acceleration = gradient = (17 − 5) m/s ÷ 8 s = 12 ÷ 8 = 1.5 m/s²
(Higher tier) distance = area = 40 m + 48 m = 88 m
(Higher tier) average speed = 88 m ÷ 8 s = 11 m/s

Answer: a = 1.5 m/s², distance = 88 m

For uniform acceleration, the average velocity is halfway between the start and end values: ½ × (5 + 17) = 11 m/s, which matches the answer above.

When the time is not given: v² − u² = 2as

For uniform acceleration you can link initial velocity u, final velocity v, acceleration a and distance s without knowing the time:

v² − u² = 2 a s (v, u in m/s; a in m/s²; s in m)

Worked example: a car overtaking

A car speeds up from 10 m/s to 30 m/s with an acceleration of 2.5 m/s². How far does it travel while accelerating?
s = (v² − u²) ÷ 2a = (30² − 10²) ÷ (2 × 2.5) = (900 − 100) ÷ 5.0

Answer: 160 m

(Higher tier) Check with the graph method: the time taken is (30 − 10) ÷ 2.5 = 8.0 s, so the area = ½ × (10 + 30) × 8.0 = 160 m. Both methods agree.

Describing and drawing journeys

The same journey can be shown on both types of graph. A pupil walks at 2 m/s for 20 s, waits for 10 s, then runs at 4 m/s for 10 s:

• When describing a graph, give the type of motion for each section and use numbers: “accelerates uniformly from 0 to 6 m/s in the first 10 s”.
• When plotting a graph from a table: label both axes with units, choose scales that use more than half the grid, plot points as small neat crosses, and join straight-line sections with a ruler.

Exam tip: Before reading any motion graph, check the y-axis: distance or velocity? The gradients mean different things – speed on a distance–time graph, acceleration on a velocity–time graph.

Common mistakes

• Reading one point and dividing (e.g. 48 m ÷ 11 s) instead of using the change in distance ÷ the change in time from a tangent.
• Drawing a “tangent” that cuts across the curve, or a very short one that is hard to read.
• Using the gradient of a velocity–time graph as the speed – it is the acceleration.
• Forgetting the rectangle under a triangle when the object does not start from rest.
• Finding average speed by averaging the speeds of different sections instead of total distance ÷ total time.

Revision questions

Try these, then check your answers below each one.

Question 1. A distance–time graph is a straight line from (0 s, 0 m) to (25 s, 200 m). Calculate the speed.

Answer: 8 m/s

Working: 200 m ÷ 25 s = 8 m/s.

Question 2. (Higher tier) A tangent is drawn to a curved distance–time graph at t = 7 s. The tangent passes through (2 s, 0 m) and (12 s, 90 m). What is the speed at t = 7 s?

Answer: 9 m/s

Working: gradient = (90 − 0) ÷ (12 − 2) = 90 ÷ 10 = 9 m/s.

Question 3. A distance–time graph is a curve that gets steeper as time goes on. What does this show? (Choose from: The object is accelerating, The object is decelerating, The object is moving at constant speed, The object is stationary.)

Answer: The object is accelerating

Working: The gradient increases, so the speed increases: the object is accelerating.

Related lessons

• Previous lesson: 9.3 More About Velocity–Time Graphs
• Next lesson: 10.1 Force and Acceleration
• In this chapter: 9.1 Speed and Distance–Time Graphs
• In this chapter: 9.2 Velocity and Acceleration

Continue in the full course

This lesson is section 9.4 (AGCPH0904) of our free GCSE Physics course for AQA. In the full course you get every video for this section, a self-marking practice quiz, a printable worksheet with a mark scheme, chapter tests, and your progress saved. Go to the full GCSE Physics course.