A- Level IAL Unit 1 05 Graphs of Motion

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Unit 1 05 Graphs of Motion

1. Learning Objectives

  • Subject Content:
    • Interpret Displacement-time () graphs: Gradient = Velocity.
    • Interpret Velocity-time () graphs: Gradient = Acceleration; Area = Displacement.
    • Interpret Acceleration-time () graphs: Area = Change in velocity.
    • Translate between different types of graphs (e.g., sketch from ).
  • Language Goals:
    • Use descriptive verbs: “Accelerating”, “Decelerating”, “Stationary”, “Constant velocity”.
    • Explain the physical meaning of “Gradient” and “Area under the graph”.

2. Key Terminology

Term Physical Meaning
Displacement-time graph Shows position change.
Velocity-time graph Shows speed/direction change.
Gradient / Slope Rate of change ().
Intercept Initial value ().
Area under the graph Product of axes ().
Stationary / At rest Velocity = 0.
Uniform / Constant Value does not change.

3. Displacement-Time Graphs ()

The graph tells us where an object is at any given time.

Key Rule:

The Gradient of an graph = Velocity ()

  • Horizontal line: Gradient is 0 Velocity is 0 (Object is Stationary).
  • Straight diagonal line: Gradient is constant Velocity is constant (Uniform Velocity).
  • Curved line: Gradient is changing Velocity is changing (Accelerating).
    • Curve getting steeper: Accelerating.
    • Curve getting flatter: Decelerating.

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Tip: If the gradient is negative (sloping down), the object is moving backwards (returning to start).

4. Velocity-Time Graphs ()

The graph is the most useful graph in A-level Physics because it links displacement and acceleration.

Key Rules:

  1. The Gradient = Acceleration ()

    • Steep slope = High acceleration.
    • Horizontal line = Zero acceleration (Constant Velocity).

  2. The Area under the graph = Displacement ()

    • Area above the t-axis is positive displacement (moving forward).
    • Area below the t-axis is negative displacement (moving backward).

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Exam Warning: Always check if the y-axis is Velocity or Speed.

  • Velocity-time graphs can go below the x-axis (negative direction).
  • Speed-time graphs can never be negative.

5. Acceleration-Time Graphs ()

These are generally simpler in Unit 2 (usually horizontal lines for constant acceleration).

Key Rules:

  1. Horizontal line: Constant Acceleration.
  2. Area under the graph = Change in Velocity ()

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6. The “Golden Cycle” of Motion Graphs

To help students remember, we can summarize the chain of relationships between derivatives and integrals:

  • Moving forward (Derivation): Look at the Gradient
  • Moving backwards (Integration): Look at the Area

7. Practice Exercises

Q1. The displacement-time graph of an object is a straight line passing through the origin with a positive gradient. This indicates the object is:

A. Stationary

B. Moving with constant velocity

C. Moving with constant acceleration

D. Moving with increasing acceleration

Q2. An object starts from rest and accelerates at for , then travels at constant velocity for .

  • (a) Sketch the velocity-time graph for this motion.
  • (b) Calculate the total displacement using the area method.

Q3. [Graph Interpretation] The velocity-time graph of a car shows it moving at , then the line slopes down, crossing the time axis, and ends at .

  • What is the physical meaning of the point where the line crosses the time axis ()?

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Q4. A ball is dropped from a height and bounces back. Which of the following graphs best represents its motion (taking downward as positive)?

(Use your imagination for the options, focus on the logic: Positive velocity sudden change to Negative velocity upon impact).

Answers

Q1. Interpreting an graph

  • Analysis:
    • The question describes a straight line displacement-time () graph passing through the origin, and the gradient is positive.
    • According to the rule: The Gradient of an graph = Velocity.
    • Since it is a straight line, it means the gradient is constant.
    • Since the gradient is positive, it means the velocity is positive.
    • Therefore, the object is moving with a constant positive velocity.
  • Correct Answer: B. Moving with constant velocity

Q2. Sketching and Calculating from a v-t graph

Description: Starts from rest, accelerates at for , then travels at constant velocity for .

(a) Sketch the velocity-time graph:

  1. Stage 1 (0-5s): Uniform acceleration.

    • Starting point: (starts from rest).
    • Because acceleration is the gradient of the graph, this is a straight line starting from the origin with a gradient of .
    • At , the velocity .
    • Drawing: A straight line connecting the point to the point .

  2. Stage 2 (5-10s): Uniform velocity.

    • The velocity remains constant at .
    • The acceleration is (gradient is ).
    • Drawing: A horizontal straight line from the point to the point .

(b) Calculate the total displacement:

  • Method: Area under the graph.
  • The graph is a trapezoid, or it can be seen as a triangle plus a rectangle.
  • Area of the triangle (0-5s): .
  • Area of the rectangle (5-10s): .
  • Total displacement: .

Q3. Physical meaning of crossing the t-axis ()

  • Analysis:
    • The graph shows the velocity changing from a positive value () to a negative value ().
    • A positive velocity means moving in one direction (e.g., forward/upward).
    • A negative velocity means moving in the opposite direction (e.g., backward/downward).
    • When the graph crosses the time axis, the velocity .
  • Answer: This point is the Turning Point of the object. This indicates that the object is Instantaneously Stationary at this moment, and immediately changes its direction of motion. (For example: the moment a ball thrown upward reaches its highest point, or the moment right before a car reverses.)

Q4. Bouncing Ball Graph (Downward positive)

  • Analysis:
  1. Drop: The velocity starts from 0 and increases downwards. This should be a straight line starting from the origin with a positive gradient (gradient = acceleration due to gravity ).
  2. Bounce: During the very short time of contact with the ground, the direction of velocity changes instantaneously. It instantly changes from the maximum positive velocity (hitting the ground downwards) to the maximum negative velocity (bouncing upwards). On the graph, this appears as an almost vertical downward line segment that crosses the time axis.
  3. Rebound upwards: The object moves upwards, and the velocity is negative. But under the effect of gravity, the upward velocity is decreasing (the value tends towards 0). The direction of acceleration is still downwards (positive direction), so the gradient of the graph is still the positive . This is a straight line pointing from the negative axis towards the time axis, parallel to the first line.
  • Conclusion: The graph that fits this process is the “sawtooth” graph shown in the previous lesson.

Consultant’s Note:

These questions cover the core concepts of motion graphs.

  • Q2 tests the ability to convert text descriptions into graphs, and the basic skill of using area to calculate displacement.
  • Q3 and Q4 test the profound understanding of the physical meaning of the positive and negative signs of velocity (directionality). This is one of the biggest differences between A-level and IGCSE/Junior high school physics.

💡 Teacher’s Guide

Special note for Q4 (Bouncing Ball):

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This is the most classic “trap” in A-level.

  • Falling process: Velocity is downwards (positive) and increases uniformly (the gradient of the line is ).
  • Moment of bounce: Velocity instantly becomes upwards (negative). On the graph, this appears as an almost vertical line jumping from a positive to a negative .
  • Rising process: Velocity is upwards (negative), but decreasing (value gets smaller, returning to zero). The gradient is still (the direction of gravitational acceleration remains unchanged).