9.1 Speed and Distance–Time Graphs

A free GCSE Physics lesson for AQA (8463) from OBA School: calculating speed, typical everyday speeds, and reading the gradient of a distance–time graph. 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

• Use speed = distance ÷ time, and rearrange it to find distance or time
• Convert minutes, hours and kilometres into seconds and metres before calculating
• Recall typical speeds for walking, running, cycling and sound in air
• Use a distance–time graph to tell whether an object is stationary or moving at constant speed
• Work out speed from the gradient of a distance–time graph, and calculate average speed

Key words

• Speed: How far an object travels each second. Speed is a scalar: it has size but no direction.
• Distance: How far an object travels, in metres (m). A scalar quantity.
• Constant speed: The same distance is covered every second (also called uniform speed).
• Average speed: Total distance travelled ÷ total time taken, for a journey where the speed changes.
• Distance–time graph: A graph of distance travelled (y-axis) against time (x-axis).
• Gradient: The steepness of a line: change in the y-value ÷ change in the x-value.

Calculating speed

The speed of an object tells you how much distance it covers every second. For an object moving at constant speed:

speed, v (m/s) = distance travelled, s (m) ÷ time taken, t (s) v = s ÷ t

Rearranged: s = v × t and t = s ÷ v

The SI unit of speed is the metre per second, m/s. Physicists use s for distance, so do not mix it up with the unit s (seconds).

Worked example: a cyclist on a cycle path

A cyclist covers 1500 m in 250 s. Calculate her speed.
v = s ÷ t = 1500 m ÷ 250 s

Answer: 6.0 m/s

Worked example: finding a distance (convert minutes first)

A train travels at a steady 45 m/s for 2.0 minutes. How far does it go?
t = 2.0 × 60 = 120 s
s = v × t = 45 m/s × 120 s = 5400 m

Answer: 5400 m (= 5.4 km)

Worked example: finding a time

Sound travels at about 330 m/s in air. How long does it take to travel 1.65 km?
s = 1.65 × 1000 = 1650 m; t = s ÷ v = 1650 ÷ 330

Answer: 5.0 s

Exam tip: Always convert to SI units before you substitute: km → m (× 1000), minutes → s (× 60), hours → s (× 3600).

Average speed and typical speeds

Real objects rarely move at a perfectly steady speed: a bus speeds up, slows down and stops. The speed equation then gives the average speed for the journey:

average speed = total distance travelled ÷ total time taken

Worked example: a bus journey

A bus travels 12 km across a city in 25 minutes, including stops.
s = 12 000 m, t = 25 × 60 = 1500 s
average speed = 12 000 ÷ 1500

Answer: 8.0 m/s

You should know some typical speeds. A person walks at about 1.5 m/s, runs at about 3 m/s and cycles at about 6 m/s. Sound travels through air at about 330 m/s. The speed of a person depends on their age, fitness, the terrain (uphill, mud, crowds) and how far they have to go.

Wind speed also varies a lot, from a gentle breeze of a few m/s up to more than 30 m/s in a storm.

Measuring speed

To measure the speed of a person or toy car, measure a distance with a tape measure or trundle wheel and time the motion with a stopwatch. Then use v = s ÷ t.

• Your reaction time (about 0.2–0.3 s) adds an error at the start and at the end of the timing. Timing over a longer distance makes this error a smaller fraction of the time measured.
• Repeat the timing and calculate a mean to reduce the effect of random errors.
• For short, fast events, light gates connected to a data logger measure the time far more precisely than a hand-held stopwatch.

Distance–time graphs

A distance–time graph shows how far an object has travelled at each moment.

• A horizontal line means the distance is not changing, so the object is stationary.
• A straight line sloping upwards means the object moves at constant speed.
• A steeper line means a greater speed.

Key fact: The gradient (slope) of a distance–time graph represents the speed. gradient = change in distance ÷ change in time

Worked example: reading the walk graph

Section AB: gradient = 90 m ÷ 60 s = 1.5 m/s (or 150 m ÷ 100 s = 1.5 m/s).
Section BC: horizontal, so the walker was stopped for 160 − 100 = 60 s.
Section CD: speed = (330 − 150) m ÷ (220 − 160) s = 180 ÷ 60 = 3.0 m/s. CD is steeper than AB.
Average speed for the whole trip = 330 m ÷ 220 s

Answer: average speed = 1.5 m/s

Exam tip: Draw the gradient triangle as large as you can – it makes your reading of the graph more accurate. Read values from grid lines where possible.

When the speed changes

If the speed of an object is changing, its distance–time graph is a curve. A curve that gets steeper shows an object speeding up; a curve that gets less steep shows an object slowing down.

(Higher tier) The speed at one instant is found by drawing a tangent to the curve at that time and working out the gradient of the tangent. You will practise this in section 9.4.

Common mistakes

• Forgetting to convert minutes or hours into seconds, or km into m, before using v = s ÷ t.
• Calculating the average speed by averaging the speeds of each section. Always use total distance ÷ total time.
• Thinking a horizontal line on a distance–time graph means constant speed – it means the object is not moving.
• Using a tiny gradient triangle, or reading the values of a single point, instead of a change in distance ÷ change in time.
• Writing the unit of speed as “m/s²” or “mps” – the unit is m/s.

Revision questions

Try these, then check your answers below each one.

Question 1. A distance–time graph for a toy train is a horizontal straight line. What does this show? (Choose from: The train is stationary, The train is moving at a constant speed, The train is speeding up, The train is slowing down.)

Answer: The train is stationary

Working: A horizontal line means the distance travelled is not changing, so the train is not moving: its speed is zero.

Question 2. A runner completes 400 m in 80 s. Calculate her average speed.

Answer: 5 m/s

Working: v = s ÷ t = 400 m ÷ 80 s = 5 m/s.

Question 3. A car travels at a constant speed of 25 m/s for 3.0 minutes. Calculate the distance it travels in metres.

Answer: 4500 m

Working: t = 3.0 × 60 = 180 s. s = v × t = 25 m/s × 180 s = 4500 m (4.5 km).

Related lessons

• Previous lesson: 8.9 Resolution of Forces
• Next lesson: 9.2 Velocity and Acceleration
• In this chapter: 9.3 More About Velocity–Time Graphs
• In this chapter: 9.4 Analysing Motion Graphs

Continue in the full course

This lesson is section 9.1 (AGCPH0901) 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.