Motion Graphs Explorer
Build a journey leg by leg, then watch the distance–time, displacement–time and velocity–time graphs draw themselves as the object moves — and read speed, acceleration and distance straight off them.
On a distance-time graph the slope gives the speed: a horizontal line means the object is at rest, and a straight slanting line means uniform speed. On a velocity-time graph the slope gives the acceleration and the area under the line gives the displacement, so an object at 20 m/s for 5 s covers 20 × 5 = 100 m.
Build the journey
Add one leg at a time. The graphs redraw instantly.
The journey
Watch the object move along the track.
Total length of the path — it can never go down.
Straight-line change in position, with a direction (+ or −).
The three graphs
All three share one time axis, so you can read across them.
Reading the graphs
The three exam skills, worked out for the leg the object is on right now.
Leg by leg
Area under the v–t graph for each leg — the working your exam answer needs.
| Leg | Motion | u (m/s) | v (m/s) | a (m/s²) | t (s) | Area under v–t | Distance (m) |
|---|
The graph shapes you must recognise
Every motion question in the exam is one of these five shapes, or a few of them joined together. Press Load to run any of them in the simulator.
Match the graph
10 questions: sometimes you get a graph and have to say what the motion is, sometimes you get the motion and have to pick the right graph.
Review
Equations of motion
Fill in any three values and leave the rest blank.
The three equations
The last one is the area of the trapezium under the v–t graph — handy when you do not know a.
Graphs are a picture of the motion
Once you can read a slope and an area, every motion graph question in Class 9, 10 and 11 becomes the same two steps.
Build the journey
Move, stand still, speed up, slow down or come back — in any order.
Press play
The object moves and all three graphs draw themselves with a marker on each.
Read the graphs
Slope of d–t = speed, slope of v–t = acceleration, area under v–t = displacement.
Practise
Match graphs to motions, then solve numericals with the equations of motion.
Distance–time graph
The slope (gradient) of a distance–time graph gives the speed.
speed = Δd ÷ Δt = (d₂ − d₁) ÷ (t₂ − t₁)
- Horizontal line → the object is at rest
- Straight slanting line → uniform speed
- Curve bending upwards → speeding up (acceleration)
- It can never slope downwards, because distance never decreases
Velocity–time graph
The slope gives the acceleration, and the area between the line and the time axis gives the displacement.
a = Δv ÷ Δt area = displacement
- Horizontal line above the axis → uniform velocity
- Straight line going up → uniform acceleration
- Straight line coming down to the axis → uniform retardation
- Below the axis → moving backwards
Distance vs displacement
Walk 40 m east and 40 m back west. You travelled a distance of 80 m, but your displacement is 0 m.
- Distance is a scalar — only size, never negative, never decreases
- Displacement is a vector — size and direction, can be negative or zero
- Distance ≥ |displacement|, always
- They are equal only when the motion stays in one straight line, one direction
Area under the v–t graph
Split the graph into rectangles and trapeziums and add them up:
rectangle = v × t triangle = ½ × t × v trapezium = ½(u + v) × t
Area below the time axis counts as negative displacement, but it still adds to the distance travelled.
The equations of motion
For uniform acceleration only:
v = u + at s = ut + ½at² v² = u² + 2as
Each one comes straight from the v–t graph: the first from its slope, the second and third from its area.
Speed or velocity?
Speed = distance ÷ time (scalar). Velocity = displacement ÷ time (vector).
A cyclist who rides one lap of a circular track and comes back to the start has a good average speed but an average velocity of zero.
average speed = total distance ÷ total time
Motion Graphs Explorer — your questions
What does the slope of a distance-time graph represent?
The slope gives the speed: speed = (d₂ − d₁) ÷ (t₂ − t₁). A horizontal line means the object is at rest, a straight slanting line means uniform speed, and a curve bending upwards means the object is speeding up. The steeper the line, the faster the motion.
What does the area under a velocity-time graph give?
The area between the line and the time axis gives the displacement. Split it into rectangles and triangles: a rectangle is v × t, a triangle is ½ × base × height, and a trapezium is ½(u + v) × t. Area below the time axis counts as negative displacement but still adds to the distance travelled.
What is the difference between distance and displacement?
Distance is the total length of the path travelled; displacement is the straight-line change in position with a direction. Walk 40 m east and 40 m back west and your distance is 80 m but your displacement is zero. Distance is a scalar and never decreases; displacement is a vector and can be negative or zero.
What are the three equations of motion?
For uniformly accelerated motion: v = u + at, s = ut + ½at² and v² = u² + 2as. Each comes straight from the velocity-time graph — the first from its slope, the other two from the area beneath it. A fourth form, s = (u + v)t ÷ 2, is the area of the trapezium and is useful when a is unknown.
Can a distance-time graph slope downwards?
No. Distance is the total path covered, so it can stay the same but can never decrease, and the graph can never go down. A displacement-time graph can slope downwards, because displacement decreases when the object moves back towards its starting point and becomes negative if it goes past it.
What does a velocity-time graph look like for uniform retardation?
It is a straight line sloping downwards towards the time axis. The slope is negative, which means the acceleration is negative — this is retardation or deceleration. A car braking from 20 m/s to rest in 4 s has a = (0 − 20) ÷ 4 = −5 m/s², and the triangular area shows it covered ½ × 4 × 20 = 40 m.
What is the difference between speed and velocity?
Speed is distance divided by time and is a scalar, so it has size only. Velocity is displacement divided by time and is a vector, so it has direction too. A cyclist who completes one lap of a circular track and returns to the start has a good average speed but an average velocity of exactly zero.