11.2 Linear Motion Graphs
Learn to interpret displacement-time and velocity-time graphs and calculate velocity, acceleration, and displacement from gradients and areas.
Learning Objectives
- State the two types of linear motion graphs: displacement-time graph and velocity-time graph.
- Explain the meaning of gradient and area under linear motion graphs.
- Interpret the motion of an object from displacement-time and velocity-time graphs.
- Solve numerical problems involving displacement-time and velocity-time graphs.
Summary of Linear Motion Graphs and Their Physical Quantities
Compare displacement-time and velocity-time graphs side by side to understand what gradients and areas represent.
Displacement-Time Graph
Gradient = Velocity
Horizontal line = Object at rest (zero velocity).
Negative gradient = Moving in the opposite direction / moving back.
Velocity-Time Graph
Gradient = Acceleration
Area under graph = Displacement
Horizontal line = Constant velocity (zero acceleration).
Short Explanation
What are linear motion graphs?
Linear motion graphs translate numerical data into visual representations. There are two types: the displacement-time graph and the velocity-time graph.
In a displacement-time graph, the vertical axis shows position and the horizontal axis shows time. The slope or gradient tells us the velocity of the object. A flat horizontal line means the position is not changing, so the object is at rest. A negative gradient means the object is travelling in the reverse direction back toward the starting point.
In a velocity-time graph, the vertical axis shows velocity. The slope or gradient indicates acceleration. A flat horizontal line does not mean the object is stationary; it means the object travels at a constant velocity with zero acceleration. To find total displacement from a velocity-time graph, calculate the total geometric area under the graph between the graph line and the time axis.
Key relationships
For a displacement-time graph: \(\text{Gradient} = \text{velocity}\).
For a velocity-time graph: \(\text{Gradient} = \text{acceleration}\) and \(\text{Area under graph} = \text{displacement}\).
Displacement-Time Graph
The gradient of a displacement-time graph represents velocity.
A displacement-time graph shows how the displacement of an object changes with time. The gradient of the graph gives the velocity of the object:
\[\text{Velocity} = \dfrac{\Delta s}{\Delta t} = \dfrac{s_2 - s_1}{t_2 - t_1}\]
| Section | Gradient | Motion of the object |
|---|---|---|
| OA | Positive, uniform | Velocity is uniform; the object moves away from the starting point with uniform speed. |
| AB | Zero (horizontal line) | Velocity is zero; the object is at rest (stationary). |
| BC | Positive, uniform | Velocity is uniform; the object continues moving forward with uniform speed. |
| CD | Negative, uniform | Velocity is negative and uniform; the object is moving in the opposite direction, turning back towards the starting point with uniform speed. |
| Curve with decreasing gradient | Decreasing | Velocity is decreasing; the object is decelerating or slowing down. |
Worked example: finding velocity
From the 3rd to the 5th second, the displacement changes from \(4\ \text{m}\) to \(8\ \text{m}\).
\[\text{Velocity} = \dfrac{(8 - 4)\ \text{m}}{(5 - 3)\ \text{s}} = \dfrac{4\ \text{m}}{2\ \text{s}} = 2\ \text{m s}^{-1}\]
From the 6th to the 12th second, the displacement changes from \(15\ \text{m}\) to \(0\ \text{m}\).
\[\text{Velocity} = \dfrac{(0 - 15)\ \text{m}}{(12 - 6)\ \text{s}} = \dfrac{-15\ \text{m}}{6\ \text{s}} = -2.5\ \text{m s}^{-1}\]
The negative sign indicates motion in the opposite direction.
Velocity-Time Graph
The gradient gives acceleration, and the area under the graph gives displacement.
A velocity-time graph shows how the velocity of an object changes with time. It allows displacement, velocity and acceleration to be determined.
\[\text{Acceleration} = \dfrac{\Delta v}{\Delta t} = \dfrac{v_2 - v_1}{t_2 - t_1}\]
\[\text{Displacement} = \text{area under the velocity-time graph}\]
| Section | Gradient | Motion of the object |
|---|---|---|
| OA | Positive, uniform | Velocity increases uniformly; acceleration is uniform. |
| AB | Zero, velocity above time axis | Velocity is constant; acceleration is zero. |
| BC | Negative, uniform | Velocity decreases uniformly; the object moves with negative acceleration or uniform deceleration. |
| CD | Zero, on the time axis | Velocity is zero; the object is stationary. Acceleration is zero. |
Worked example: finding acceleration
From the 8th to the 12th second, the velocity changes from \(3\ \text{m s}^{-1}\) to \(0\ \text{m s}^{-1}\).
\[\text{Acceleration} = \dfrac{(0 - 3)\ \text{m s}^{-1}}{(12 - 8)\ \text{s}} = \dfrac{-3\ \text{m s}^{-1}}{4\ \text{s}} = -0.75\ \text{m s}^{-2}\]
Worked example: finding displacement
In the first 4 seconds, the area under the graph is a rectangle:
\[\text{Displacement} = 4\ \text{s} \times 1\ \text{m s}^{-1} = 4\ \text{m}\]
Between 0 and 10 seconds, the area is a triangle plus a rectangle:
\[\text{Displacement} = \left[\dfrac{1}{2} \times (6\ \text{s})(15\ \text{m s}^{-1})\right] + \left[(10\ \text{s} - 6\ \text{s})(15\ \text{m s}^{-1})\right] = 45\ \text{m} + 60\ \text{m} = 105\ \text{m}\]
Try to Answer First
Answer in your mind, then press “Check Answer”.
What physical quantity is represented by the gradient of a displacement-time graph?
What does a horizontal line above the time axis represent in a velocity-time graph?
How do you determine the displacement of an object from a velocity-time graph?
Common Mistakes
- Confusing the meaning of a horizontal line: mistaking a horizontal line in a velocity-time graph for an object at rest. A horizontal line in a velocity-time graph means constant velocity, not stationary.
- Calculating the area under a displacement-time graph. The area under a displacement-time graph does not represent any meaningful physical quantity in Form 4 Science.
- Forgetting negative signs: omitting the negative sign when calculating a negative gradient, which represents negative velocity or deceleration.
- Writing incorrect units: confusing the unit of velocity, \(\text{m s}^{-1}\), with acceleration, \(\text{m s}^{-2}\).
Concept Misunderstandings
A negative gradient on a displacement-time graph means the object is decelerating.
A negative gradient on a displacement-time graph means the object has a negative velocity, indicating that it is moving in the opposite direction, returning towards the reference or starting point.
A horizontal line on any motion graph means the object has stopped moving.
On a displacement-time graph, a horizontal line means displacement is constant, so velocity is zero and the object is stationary. On a velocity-time graph, a horizontal line means velocity is maintained at a non-zero constant value, so the object is moving with zero acceleration.
Summary
- Linear motion graphs consist of displacement-time graphs and velocity-time graphs.
- For a displacement-time graph, the gradient represents velocity in \(\text{m s}^{-1}\). A horizontal line indicates that the object is at rest, and a negative gradient indicates motion in the opposite direction.
- For a velocity-time graph, the gradient represents acceleration in \(\text{m s}^{-2}\), and the area under the graph represents displacement in \(\text{m}\).
- A horizontal line on a velocity-time graph indicates constant velocity with zero acceleration, while a downward slope represents negative acceleration or deceleration.
Short Activity
A. Objective Quiz
1 What does the gradient of a velocity-time graph represent?
2 Based on a displacement-time graph, what is the state of motion of an object when the line is horizontal?
B. Fill in the Blanks
3 The displacement of an object can be determined by calculating the under a velocity-time graph.
4 A negative gradient on a displacement-time graph indicates that the object is moving in the direction.
C. Drag and Drop: Match the Graph Feature to Its Representation
Drag each graph feature to the physical quantity or motion state it represents. On a phone, tap an answer first, then tap a matching box. Tap the Answer Choices area to return a selected item.
C. Match the Graph Feature to Its Representation
Write the correct graph feature next to each representation.
| Representation | Graph Feature |
|---|---|
| 1. Velocity of the moving object | |
| 2. Acceleration of the moving object | |
| 3. Displacement travelled by the object | |
| 4. Motion with constant velocity and zero acceleration |
Answer choices: Gradient of displacement-time graph • Gradient of velocity-time graph • Area under velocity-time graph • Zero gradient on velocity-time graph
