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AQA GCSE Physics · Topic 4.5 Forces · Combined and Triple
Edexcel International GCSE Physics · Section 1 Forces and motion
Motion Formulas and Graphs
In this lesson you'll learn to: use v = s ÷ t, a = (v − u) ÷ t and
v² = u² + 2as; read a distance–time graph (gradient = speed); read a
velocity–time graph (gradient = acceleration, area = distance); and tell an
instantaneous value (a tangent) from an average one.
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Prefer to watch? Scroll to the bottom for the video on this topic
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A video walkthrough is coming soon — it will appear at the bottom of this page
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Start here — have a play
Don't read anything yet. Press play and watch the two graphs draw themselves in time with the motion. Try all four: a cyclist at a steady speed, a cliff diver falling with no air resistance, a skydiver reaching terminal velocity, and a bike journey. Then pause and use the tools — a tangent, an average, the area under the graph. The last tab is the calculations.
t = 0.0 s
s = 0 m
v = 0.0 m/s
Pause, then read the graphs:
Distance–time graph · gradient = speed
Velocity–time graph · gradient = acceleration · area = distance
Speed
v = s ÷ t
v in m/s · s in m · t in s
Acceleration
a = (v − u) ÷ t
a in m/s² · v, u in m/s · t in s
No time given
v² = u² + 2as
v, u in m/s · a in m/s² · s in m
Question 1
right first time 0 / 0
Choosing the equation. Write down what you know: s, u, v, a, t. If there is no time in the question and you don't need one, it is v² = u² + 2as. "From rest" means u = 0. "Comes to a stop" means v = 0. Convert km, minutes and ms first.
Physics With Kate · physicswithkate.com — GCSE & International GCSE Physics
▲ Drag the time slider to any moment, then switch on the tangent — the readout does the gradient for you.
Two graphs, two rules each.
Distance–time: the gradient is the speed. Velocity–time: the gradient is the acceleration and the area is the distance. Everything else on this page comes out of those four ideas and three equations.
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Revision notes
The three equations of motion
These are the three on the Motion Formulas workbook page. Learn them — they are not all given to you.
| Equation | Use it for | Units |
|---|---|---|
| v = s ÷ t average speed = distance moved ÷ time taken | speed, distance or time at a steady (or average) speed | v m/s · s m · t s |
| a = (v − u) ÷ t acceleration = change in velocity ÷ time taken | how quickly the velocity changes | a m/s² · v, u m/s · t s |
| v² = u² + 2as | uniform acceleration when no time is given | v, u m/s · a m/s² · s m |
| Symbol | Means | Unit |
|---|---|---|
| s | distance (or displacement) | m |
| t | time | s |
| u | initial (starting) velocity | m/s |
| v | final velocity — or the speed in v = s ÷ t | m/s |
| a | acceleration | m/s² |
AQA On the AQA equation sheet the last one is written
v² − u² = 2as — the same equation, rearranged. AQA writes distance = speed × time as
s = v t. Free fall near the Earth's surface: a ≈ 9.8 m/s².
AQA Typical speeds to know: walking about 1.5 m/s ·
running about 3 m/s · cycling about 6 m/s · a car in town about 13 m/s · on the
motorway about 31 m/s · a train up to 50 m/s · sound in air about 330 m/s. The speed of the
wind and of sound also vary.
⭐ Read the words, find the hidden numbers
"From rest" or "starts from stationary" means u = 0. "Comes to a stop" means v = 0. "Dropped" with no air resistance means u = 0 and a = 10 m/s². "Decelerates" means a is negative.
⚠ Trap 1 — convert before you substitute
km → m: × 1000 · minutes → s: × 60 · ms → s: ÷ 1000. A cyclist going 3.6 km in 12 minutes is going 3600 m in 720 s. More on this in the Units lesson.
⚠ Trap 2 — v² = u² + 2as gives you v², not v
Square the speeds (not 2 × them), add, then square-root at the end. If v² comes out as 400, the answer is v = 20 m/s, not 400.
Distance–time graphs
Distance on the y-axis, time on the x-axis. The gradient is the speed.
- Flat (horizontal) line — distance not changing: stationary.
- Straight sloping line — constant gradient: constant speed.
- Steeper line — bigger gradient: faster.
- Curving up (getting steeper) — speeding up (accelerating).
- Curving over (getting flatter) — slowing down.
- On a displacement–time graph, a negative gradient means it is moving back towards the start.
Instantaneous speed and average speed
On a curved distance–time graph the speed is different at every moment, so there are two different questions you can be asked:
- Instantaneous speed — the speed at one moment. Draw a tangent to the curve at that time (a straight line that just touches it), draw a big triangle on the tangent, and work out gradient = Δs ÷ Δt.
- Average speed — over a stretch of time. No tangent: just total distance ÷ total time.
Velocity–time graphs
Velocity on the y-axis, time on the x-axis. Two rules:
- The gradient is the acceleration. Sloping up = speeding up; flat = constant velocity (zero acceleration); sloping down = slowing down (negative gradient, a deceleration).
- The area under the graph is the distance travelled. Split it into rectangles and triangles. Under a curve, count the squares.
Instantaneous and average acceleration
The same idea as speed. On a curved velocity–time graph (like the skydiver):
- Instantaneous acceleration at one moment = the gradient of the tangent at that time, Δv ÷ Δt.
- Average acceleration over a stretch = total change in velocity ÷ total time, which is just a = (v − u) ÷ t.
⚠ Trap 3 — which graph are you looking at?
A flat line on a distance–time graph means stationary. A flat line on a velocity–time graph means constant velocity — still moving! Read the y-axis label before you say a word.
⚠ Trap 4 — the skydiver
Her distance–time graph increases at an increasing rate, then becomes a straight line. Her velocity–time graph increases at a decreasing rate and levels off at terminal velocity, when air resistance = weight. Don't swap the two descriptions.
How to write it in the exam
Every calculation, the same four lines
equation → substitute (units converted)
→ answer + unit
write the equation first numbers in, in m, s, m/s right sig figs, right unit
A cyclist travels 3.0 km in 10 minutes. Calculate her average speed in m/s.
v = s ÷ t · 3.0 km = 3000 m, 10 min = 600 s · v = 3000 ÷ 600 = 5.0 m/s
A car speeds up from 5.0 m/s to 20 m/s in 300 ms. Calculate its acceleration.
a = (v − u) ÷ t · 300 ms = 0.30 s · a = (20 − 5.0) ÷ 0.30 = 15 ÷ 0.30 = 50 m/s²
A train at 15 m/s accelerates uniformly at 0.50 m/s² over 0.50 km. Calculate its final velocity.
No time given → v² = u² + 2as · 0.50 km = 500 m · v² = 15² + 2 × 0.50 × 500 = 225 + 500 = 725 · v = √725 = 27 m/s (2 s.f.)
Use the velocity–time graph above to find the distance the bike travels while braking.
Distance = area under the graph from 10 s to 14 s: a triangle, ½ × base × height = ½ × 4 × 8 = 16 m
Explain how you would find the speed of an object at t = 3 s from a curved distance–time graph.
Draw a tangent to the curve at t = 3 s. Calculate its gradient = change in distance ÷ change in time, using a large triangle. The gradient is the instantaneous speed.
✅ Check your answer
Did you write the equation first? Convert km, minutes and ms? Square-root if you used v² = u² + 2as? Give the unit — m/s for speed, m/s² for acceleration, m for distance? Round to the sig figs the question asks for?
📘 Now do it in your workbook
The Motion Formulas page: write in the three equations and do the three exam-style calculations. The Motion Graphs page: read four graphs — stationary, fastest, tangents, averages and areas.
Edexcel iGCSE Forces and Motion Workbook · pages 4–5 AQA GCSE Physics Workbook · FORCES · pages 10–11✅ Can you do it? Tick as you go
0% complete🎉 Nice work! You've ticked off every objective for these spec points.
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Prefer to watch? Here's the whole thing
Video walkthrough — coming soon
Everything above, explained out loud — useful for a last-minute recap, or if you'd rather hear it than read it.
Everything above, explained out loud — useful for a last-minute recap, or if you'd rather hear it than read it. The video is on its way.
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Video coming soon
The walkthrough for this lesson is being filmed and will appear right here. Until then, the notes and the free workbook pages above cover every mark.
Now put it into practice
Pages 4 and 5 of the Forces and Motion workbook are free: three exam-style calculations and four graphs to read. The full book covers the whole topic with worked answers.
Pages 10 and 11 of the Forces workbook are free: the three equations, exam-style calculations and four graphs to read. The full book covers the whole topic with worked answers.
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