Gradient is a rate: velocity on a position graph, acceleration on a velocity graph.
Step 1: Something to Notice
Watch first. The explanation comes later.
Two position–time graphs cross. At that instant the two objects are at the same place, and one is moving three times as fast.
Which feature of the graph tells you the speed?
Step 2: Find Out
Isolate the motion type and note what the gradient and the area each represent.
Step 1 — Predict
What does the gradient of a position–time graph give?
Choose what you think will happen. You cannot see the experiment until you do — guessing first is what makes it worth watching.
Step 3: So Here Is Why
Now the explanation, after you have seen it happen.
Area is an accumulation
Velocity multiplied by time is displacement.
A curved position graph means acceleration
The gradient is changing, and the gradient is the velocity.
Area below the axis is negative displacement
Out and back gives zero displacement and plenty of distance.
A motion graph is not a map
The horizontal axis is time, whatever the line looks like.
Gradients and areas say everything
The slope of a position graph is velocity and of a velocity graph is acceleration. The area under a velocity graph is displacement and under an acceleration graph is change in velocity.
Graphs work when equations do not
Because slope and area are defined regardless of whether acceleration is constant, graphical analysis handles cases the kinematic equations cannot.
Step 4: A Common Mistake
Lots of people think
“A position–time graph shows the shape of the path traveled.”
Step 5: Why Every Instrument Trace Is a Motion Graph
The same idea somewhere new.
The gradient-and-area reading is not confined to kinematics. On a seismogram, the trace is ground velocity against time and its gradient is acceleration, which is what determines the force on a building. On an ECG the area under a segment carries clinical meaning, and on a flow meter the area under a flow-against-time trace is the total volume delivered. The skill being practiced here — asking what the gradient means and what the area means, in the units of the axes — transfers to every instrument a student will meet later, which is why it is worth more than any single kinematics problem.
Step 6: Think It Through
Practice makes it stick.
Out and Back
Problem 1 of 2
A velocity–time graph has equal areas above and below the axis. What is the displacement?
A Straight Road Over a Hill
Problem 2 of 2
A car crosses a hill on a straight road at constant speed. What shape is its position–time graph?
Slopes and Areas
1 of 5
What does the gradient of a position–time graph give?
2 of 5
What does the area under a velocity–time graph give?
3 of 5
What does a curved position–time graph mean?
4 of 5
How is area below the time axis counted?
5 of 5
How is a gradient found at a single instant on a curve?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Does a position–time graph show the shape of the path?
Step 8: Explain It in Writing
Claim, evidence, then reasoning.
The question
Explain what gradients and areas mean on each kind of motion graph.
Fill in all three boxes. The reasoning box is the one that matters most.
What You Found Out
- Gradient gives a rate; area gives an accumulation.
- A curved position–time graph means the velocity is changing.
- Area below the axis is negative displacement.
- A motion graph is not a map of the route.