Positive gradient
Motion in the chosen positive direction.
Learning Objectives
- distinguish distance from displacement;
- distinguish speed from velocity;
- interpret displacement–time graphs;
- interpret velocity–time graphs;
- connect gradient and area to physical meaning.
Introduction
This lesson develops Displacement and Velocity–Time Graphs from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
Distance and Displacement Are Not the Same
Distance measures the total path travelled. Displacement measures the change in position from the starting point and includes direction.
A learner walks 5 m east and then 5 m west. The distance is 10 m, but
the displacement is 0 m.
| Quantity | Includes direction? | Can be negative? |
|---|---|---|
| Distance | No | No |
| Displacement | Yes | Yes |
| Speed | No | No |
| Velocity | Yes | Yes |
🇿🇼 Zimbabwe example: The same graph skills can model mobile-data use, ZESA electricity units, rainfall in Mutare, kombi journeys and school attendance.
Displacement–Time Graphs
The gradient of a displacement–time graph gives velocity.
Negative gradient
Motion in the opposite direction.
Horizontal section
No change in position: stationary.
Curved section
Velocity is changing.
Velocity–Time Graphs
A velocity–time graph can extend below the time axis. Negative velocity means movement in the opposite direction.
- Gradient gives acceleration.
- Signed area gives displacement.
- Area above the axis is positive.
- Area below the axis is negative.
net displacement = positive area − magnitude of negative area
Worked Example
A velocity–time graph forms a triangle above the axis with base 6 s and height 8 m/s.
1
Displacement equals area.
2
Area = ½ × 6 × 8 = 24 m.
3
The positive sign means displacement in the chosen positive direction.
⚠️ Common Mistake
On a velocity–time graph, a line below the axis does not mean negative
speed. It represents velocity in the opposite direction.
✅ Quick Check
Can displacement be zero when distance is not zero?
What does gradient mean on a displacement–time graph?
What does area mean on a velocity–time graph?
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- Explain the difference between distance and displacement.
- State the meaning of a negative velocity.
- Find the displacement represented by a rectangle 5 s wide and 6 m/s high.
- A graph has +30 m area above the axis and −12 m area below. Find net displacement.
- A displacement changes from −4 m to 8 m in 6 s. Find average velocity.
- Describe a journey with zero final displacement but non-zero total distance.
✅ Check Your Work
Attempt the questions before opening the solutions.
Show answers and working
Foundation
- Distance is total path length; displacement is change in position with direction.
- The object is moving in the chosen negative direction.
- Displacement=5×6=30 m.
Developing
- Net displacement=30−12=18 m.
- Change in displacement=8−(−4)=12 m; average velocity=12÷6=2 m/s.
Challenge
- Travel 10 m east and then 10 m west. Final displacement is 0, but distance is 20 m.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Displacement and velocity include direction.
- Gradient of displacement–time gives velocity.
- Gradient of velocity–time gives acceleration.
- Signed area under velocity–time gives displacement.