2.2 Linear Motion Graphs
Learn how to interpret, analyse, and sketch displacement-time, velocity-time, and acceleration-time graphs to determine physical quantities of linear motion.
Learning Objectives
- Interpret displacement-time, velocity-time, and acceleration-time graphs for different types of linear motion.
- Determine distance, displacement, speed, and velocity from a displacement-time graph.
- Determine distance, displacement, velocity, and acceleration from a velocity-time graph.
- Sketch and convert between displacement-time, velocity-time, and acceleration-time graphs.
- Solve quantitative and qualitative problems involving linear motion graphs.
Summary of Motion Graphs Relationships
Interpreting gradient and area to connect displacement, velocity and acceleration.
From Graph to Quantity
The gradient of a displacement-time graph gives velocity. The gradient of a velocity-time graph gives acceleration. The area under a velocity-time graph gives displacement.
Key relationships
\(v = \dfrac{\Delta s}{\Delta t}\) \(a = \dfrac{\Delta v}{\Delta t}\) \(s = \text{area under } v-t \text{ graph}\)Graph Conversion Flow
Displacement-time graph \(\xrightarrow{\text{gradient}}\) Velocity-time graph \(\xrightarrow{\text{gradient}}\) Acceleration-time graph.
Reverse direction: the area under an acceleration-time graph gives the change in velocity \(\Delta v\). Combine it with the initial velocity \(v_0\) to determine velocity. Then, the area under a velocity-time graph gives the change in displacement \(\Delta s\), which is combined with the initial displacement \(s_0\) to determine absolute displacement.
Short Explanation
How to read and use the three main linear motion graphs.
Displacement-Time Graph
The gradient gives velocity. A horizontal line means the object is at rest. A straight diagonal line means uniform velocity. A curve curving upward means acceleration.
Velocity-Time Graph
The gradient gives acceleration. The area under the graph gives displacement. A horizontal line means uniform velocity and zero acceleration.
Acceleration-Time Graph
A line on the time axis means zero acceleration. A horizontal line above or below the axis means uniform acceleration. An upward-sloping line means increasing acceleration, while a downward-sloping line means decreasing acceleration.
Direction Matters
Areas above the time axis represent positive displacement. Areas below the time axis represent negative displacement. For total distance, treat all areas as positive.
| Graph | Gradient gives | Area under graph gives | Horizontal line means |
|---|---|---|---|
| Displacement-time | Velocity | No physical significance in this syllabus | At rest |
| Velocity-time | Acceleration | Displacement | Uniform velocity |
| Acceleration-time | Not required | Change in velocity | Zero acceleration |
Try to Answer First
Answer in your mind, then press “Check Answer”.
What does the gradient of a displacement-time graph represent?
Check Answer
How can displacement be determined from a velocity-time graph?
Check Answer
What does a horizontal line on a velocity-time graph indicate about the acceleration of the object?
Check Answer
Common Mistakes
- Confusing the area under a displacement-time graph with displacement. Displacement is read directly from the vertical axis on an \(s-t\) graph.
- Adding areas below the time axis with a positive sign when calculating displacement. Areas below the axis represent negative displacement and must be subtracted.
- Forgetting that a negative gradient on a displacement-time graph means moving in the opposite direction, not deceleration.
Concept Misunderstandings
A downward-sloping straight line on a displacement-time graph means the object is decelerating.
A downward straight line on an \(s-t\) graph has a constant negative gradient, which means the object is moving at a uniform velocity in the negative direction.
Distance and displacement obtained from a velocity-time graph are always the same.
Distance is the total area without regard to sign, whereas displacement considers direction where areas below the axis are subtracted.
Summary
- Displacement-time graph: Gradient = velocity. Horizontal line = at rest (\(v = 0\)).
- Velocity-time graph: Gradient = acceleration. Area under graph = displacement. Horizontal line = uniform velocity (\(a = 0\)).
- Acceleration-time graph: Horizontal line at non-zero value = uniform acceleration. Line on time axis = zero acceleration.
- Average speed = \(\dfrac{\text{Total distance}}{\text{Total time}}\).
- Average velocity = \(\dfrac{\text{Total displacement}}{\text{Total time}}\).
Short Activity
Objective: Interpret motion graphs and identify the physical quantities they represent.
A. Objective Quiz
1 Which physical quantity is represented by the gradient of a velocity-time graph?
2 Based on a velocity-time graph, what does the area under the graph representing a region below the time axis indicate?
B. Fill in the Blanks
3 The gradient of a displacement-time graph gives the of the object.
4 On a velocity-time graph, a horizontal straight line with zero gradient indicates that the object has acceleration.
C. Matching / Drag and Drop
Drag each graphical feature to the correct meaning. If using a phone, tap the answer first, then tap the matching box.
