Drag points to shape a distance-time or velocity-time graph and a runner moves to match, with the six graph shapes, the two graphs compared side by side, and a worked area calculation.
On a distance-time graph the SLOPE is the speed: steeper means faster, flat means stopped, and a downward slope means moving back toward the start.
Drag the dots (or focus one and use the arrow keys) to reshape the motion.
What a motion graph is
A motion graph is a graph with time along the bottom that describes how something
moves. Two kinds cover almost everything asked of them:
A distance-time graph plots how far the object is from where it started.
A velocity-time graph plots how fast it is going.
Between them they hold the whole story of a journey: where it is, how fast, which way,
and whether it is speeding up. The trick is that the same drawn line means two
different things depending on which of the two graphs you are looking at, and most of
the difficulty in this topic is that one fact.
Reading motion from a graph
The interactive above lets you drag the points to shape either graph, and a runner
moves along a track to match. Press Play to watch the motion trace out in real time.
The Shape buttons load the six standard shapes straight in, so you can see what each
one does to the runner rather than fighting four control points into position. The key to
reading any motion graph is the same idea you meet in math: the slope of the
line.
The six shapes and what each one means
Almost every motion-graph question is really a shape-matching question. There are six
shapes worth knowing, and each one says something different depending on which graph it
is drawn on.
Six shapes a motion graph can take, with what each one means on a distance-time
graph and on a velocity-time graph.
Shape
On a distance-time graph
On a velocity-time graph
Horizontal line
Stopped. The distance is not changing, so the speed is zero, even though time keeps passing.
Constant velocity. The velocity is not changing, so the acceleration is zero and the runner keeps the same speed.
Straight line sloping up gently
Moving away from the start at a slow, steady speed. The slope is small, so the speed is small.
Speeding up gently. The slope is a small positive acceleration, so the velocity climbs slowly.
Straight line sloping up steeply
Moving away from the start fast. Same kind of motion as the gentle line, but the steeper slope means the higher speed.
Speeding up hard. The steeper slope is the larger acceleration.
Straight line sloping down
Moving back toward the start. The distance is falling, so the slope is negative and the direction has reversed.
Slowing down at a steady rate. The velocity is falling, which is a negative acceleration.
Curve getting steeper
Speeding up. Each section is steeper than the last, so the speed is rising: this is what acceleration looks like on a distance-time graph.
The acceleration itself is growing, so the velocity climbs faster and faster.
Curve levelling off
Slowing down. Each section is less steep than the last, so the speed is falling and the runner is coming to rest.
Still speeding up, but by less and less. The acceleration is dropping toward zero.
Load any of these with the Shape buttons in the tool, switch between
Distance-time and Velocity-time, and watch the runner do two completely different
things with the same line.
Distance-time graphs: slope is speed
On a distance-time graph, time runs along the bottom and distance up the side. The
slope of the line is the speed:
A steep line means fast.
A gentle line means slow.
A flat (horizontal) line means stopped.
A line sloping down means moving back toward the start.
Slope is rise over run. Here the rise is a distance and the run is a time, so the slope is
distance divided by time, which is exactly what speed is. That is the whole reason a
steeper line is a faster one: in the same amount of time it covers more ground.
Direction is the sign of that slope. A line rising to the right means moving away from
the start; a line falling to the right means coming back, because the distance from the
start is shrinking. Nothing else on the graph tells you the direction.
Which graph shows the fastest runner?
It depends which graph you are given, and the two answers are not the same:
On a distance-time graph, the fastest runner is the steepest line.
On a velocity-time graph, the fastest runner is the highest line.
On a velocity-time graph the height of the line is the velocity, so a line up near the
top is fast whether it is climbing, falling or dead flat. The slope of that graph is not
the speed at all: it is the acceleration. Carrying “steep means fast” across from the
first graph to the second is the single most common mistake in this topic, and it turns
“slowing down from a high speed” into “going fast”, which are opposite answers.
Distance-time or velocity-time?
Distance-time graphs and velocity-time graphs compared: what the axis shows,
how to spot fast motion, and what the slope, area, horizontal lines and
downward lines mean on each.
Distance-time graph
Velocity-time graph
What the y-axis shows
How far the object is from the start, in metres.
How fast the object is going, in metres per second.
How you spot fast motion
A steep line. Fast is about the slope, not the height.
A high line. Fast is about the height, not the slope.
What the slope means
The speed.
The acceleration.
What the area under the line means
Nothing useful. Distance times time is not a quantity.
The distance travelled.
What a horizontal line means
Stopped.
Moving at a constant velocity, not stopped.
What a downward line means
Coming back toward the start.
Slowing down.
What a point on the axis (y = 0) means
At the starting point.
At rest, not moving at all.
Velocity-time graphs: slope is acceleration, area is distance
Switch the graph to Velocity-time. Now the height of the line is the velocity itself,
and two new ideas appear:
The slope of the line is the acceleration (how quickly the velocity changes).
Sloping up is speeding up; sloping down is slowing down; flat is constant velocity.
The area under the line is the distance travelled. A constant velocity traces a
rectangle (velocity times time); a changing velocity traces a triangle or trapezium.
Finding the distance from the area
Take a runner who accelerates from rest to 6 m/s over 3 seconds, holds that speed for 3
seconds, then eases off to 2 m/s by 10 seconds. How far did they go?
Break the region under the line into shapes you already know, work out each area, and add
them up.
Each shaded region is a distance. Add them up and you have the whole journey.
The area under each section of the velocity-time graph, the shape it makes, the
arithmetic and the distance it represents.
Section
Shape
Area
Distance
0 s to 3 s
triangle
½ × 3 s × 6 m/s
9 m
3 s to 6 s
rectangle
6 m/s × 3 s
18 m
6 s to 10 s
trapezium
(2 m/s × 4 s) + (½ × 4 s × 4 m/s)
16 m
Whole journey
9 + 18 + 16
43 m
Two things are worth noticing. The last section is a trapezium, and the safe way to
handle it is as a rectangle sitting under the lower of its two ends plus a triangle on
top: 2 m/s times 4 s, plus half of 4 s times the 4 m/s it drops. And the runner is
slowing down across that last section but still covering 16 m, more than the 9 m
covered while speeding up. Slowing down is not stopping.
Speed and velocity: what is actually moving
Speed is how fast something moves (for example 5 m/s). Velocity is speed plus a
direction. On a graph, a positive slope and a negative slope are both real motion, just
in opposite directions, which is why velocity, not just speed, is what physics tracks.
Why this connects to math
“Slope = speed” and “slope = acceleration” are the same slope you compute
on a coordinate plane: rise over run. Motion graphs are just a
coordinate plane where the x-axis is time. Once you can read a slope, you can read motion.
Using this with a class
Call out a motion in words (“she walks fast, stops to tie a shoe, then jogs home”) and
have students build the distance-time graph to match, then check with Play. Or show one
of the six shapes, ask what it means, then switch the graph type without changing the line
and ask again. It is free to embed on your own site or LMS.
Frequently asked questions
What does the slope of a distance-time graph represent?
The slope of a distance-time graph is the speed. A steep slope means fast motion, a gentle slope means slow motion, a flat (horizontal) line means the object is stopped, and a downward slope means it is moving back toward the start. Slope is rise over run, which here is distance divided by time, and distance over time is exactly speed.
What does the slope of a velocity-time graph represent?
The slope of a velocity-time graph is the acceleration. A line sloping up means speeding up (positive acceleration), a line sloping down means slowing down, and a flat line means constant velocity (zero acceleration). Slope here is change in velocity divided by time, which is the definition of acceleration.
What does the area under a velocity-time graph represent?
The area between a velocity-time line and the time axis is the distance travelled (displacement). For a constant velocity the area is a rectangle (velocity times time); for changing velocity it is a triangle or trapezoid. This works because distance equals velocity times time, and that product is exactly the area under the line.
What is the difference between speed and velocity?
Speed is how fast something moves (a scalar, just a number with a unit like m/s). Velocity is speed together with a direction (a vector). A car going around a roundabout at a steady 30 km/h has constant speed but changing velocity, because its direction keeps changing. A change in velocity is an acceleration, so turning is a kind of acceleration even at constant speed.
How do you find acceleration from a velocity-time graph?
Find the slope of the line: pick two points, take the change in velocity (rise) and divide by the change in time (run). For example, if velocity goes from 2 m/s to 8 m/s over 3 seconds, the acceleration is (8 - 2) / 3 = 2 m/s squared. A steeper line means a larger acceleration.
Why does a steeper line on a distance-time graph mean a faster speed?
Because the slope of a distance-time graph is the speed. Slope is rise over run, and here the rise is distance and the run is time, so the slope is distance divided by time, which is the definition of speed. A steeper line covers more distance in the same amount of time, so it is the faster motion. Two runners on the same graph are easy to compare: the steeper line is always the faster runner.
How does a velocity-time graph show that a runner is moving fast?
By how HIGH the line is, not by how steep it is. On a velocity-time graph the height of the line is the velocity itself, so a line up near the top is fast motion and a line near the axis is slow motion. The slope on this graph means something else entirely: it is the acceleration. Carrying the distance-time rule (steep means fast) across to a velocity-time graph is the most common mistake in this topic.
What is the difference between a distance-time graph and a velocity-time graph?
The y-axis. On a distance-time graph it is how far from the start the object is, so the slope is the speed and a horizontal line means stopped. On a velocity-time graph it is how fast the object is going, so the slope is the acceleration, the area under the line is the distance, and a horizontal line means moving at a steady speed rather than stopped. The same drawn shape means two different things depending on which graph it is on.
How does a distance-time graph show which direction something is moving?
By whether the line goes up or down. A line rising to the right means the object is getting further from the start; a line falling to the right means it is coming back toward the start, because the distance from the start is shrinking. A horizontal line means it is not moving in either direction. Direction shows up as the sign of the slope: positive is away, negative is back.
What does a horizontal line mean on a motion graph?
It depends on the graph. On a distance-time graph a horizontal line means the object is stopped: time passes but the distance does not change. On a velocity-time graph a horizontal line means the opposite of stopped for any line above the axis, because the velocity is steady and the object keeps moving at that speed. Only a horizontal line sitting on the axis, at v = 0, means stopped.
How do you find the distance from a velocity-time graph?
Find the area between the line and the time axis. Break the shape into rectangles, triangles and trapeziums, work out each area, and add them up. A rectangle is velocity times time; a triangle is half the base times the height. It works because distance equals velocity times time, and that product is exactly what an area on these axes measures.
What is a motion graph?
A motion graph is a graph with time on the horizontal axis that describes how an object moves. The two usual kinds are the distance-time graph, which plots how far the object is from its starting point, and the velocity-time graph, which plots how fast it is going. Between them they carry the whole story of a journey: position, speed, direction and acceleration.
Which graph represents the fastest runner?
On a distance-time graph, the one with the steepest line. On a velocity-time graph, the one whose line sits highest. If several runners are drawn on the same distance-time axes, the fastest is the line that climbs most sharply, and the one that reaches a given distance first.