Kinematics Graphs and Calculus

Overview

Graphs are one of the most important tools in H2 Physics kinematics. They allow students to:

• describe motion visually
• determine instantaneous quantities
• calculate displacement or change in velocity
• translate between words, graphs, and equations

This page focuses on:

• displacement-time graphs
• velocity-time graphs
• acceleration-time graphs
• gradient meaning
• area meaning
• qualitative graph interpretation

Main topic: Kinematics

Why It Matters

Graph interpretation is where many kinematics marks are won or lost, especially when questions test motion qualitatively.

Definition

Kinematics graphs show how displacement, velocity, and acceleration vary with time, while calculus explains why gradients and areas carry physical meaning.

Core Calculus Relationships

Key Representations

For one-dimensional motion:

$$v=\frac{ds}{dt}$$ $$a=\frac{dv}{dt}$$

Hence:

• velocity is the rate of change of displacement
• acceleration is the rate of change of velocity

Reverse relationships:

$$\Delta s=\int vdt$$ $$\Delta v=\int adt$$

At H2 level, these mainly explain gradient and area ideas.

1. Displacement-Time Graphs

Meaning

Shows how displacement changes with time.

Vertical axis: displacement $s$
Horizontal axis: time $t$

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Gradient Gives Velocity

$$\text{gradient}=\frac{ds}{dt}=v$$

Figure: On a displacement-time graph, average velocity over an interval comes from the secant gradient, while instantaneous velocity at a point comes from the tangent gradient.

Interpretation

• positive gradient → moving in positive direction
• negative gradient → moving in negative direction
• zero gradient → stationary
• steeper gradient → greater speed

Straight vs Curved Graphs

Straight Line

Constant gradient → constant velocity

Curve

Changing gradient → changing velocity

Instantaneous Velocity

For a curved graph, draw a tangent at the point.

Its gradient gives instantaneous velocity.

Example

If the graph rises, becomes flat, then falls:

• object moves forward
• stops momentarily
• moves back

2. Velocity-Time Graphs

Meaning

Shows how velocity changes with time.

Vertical axis: velocity $v$
Horizontal axis: time $t$

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Gradient Gives Acceleration

$$\text{gradient}=\frac{dv}{dt}=a$$

Interpretation

• positive gradient → positive acceleration
• negative gradient → negative acceleration
• zero gradient → constant velocity

Area Gives Displacement

$$\text{area under graph}=\Delta s$$

Signed area matters:

• above axis = positive displacement
• below axis = negative displacement

Distance vs Displacement

If velocity changes sign:

• displacement = signed total area
• distance = sum of magnitudes of areas

Example

Triangle above axis area = 12 m
Triangle below axis area = 5 m

Then:

• displacement = $12-5=7$ m
• distance = $12+5=17$ m

3. Acceleration-Time Graphs

Meaning

Shows how acceleration changes with time.

Vertical axis: acceleration $a$
Horizontal axis: time $t$

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Area Gives Change in Velocity

$$\text{area under graph}=\Delta v$$

If acceleration is constant:

• horizontal line above axis → steady increase in velocity
• horizontal line below axis → steady decrease in velocity

Example

Acceleration $=3{\text{m s}}^{-2}$ for 4 s:

$$\Delta v=at=12{\text{m s}}^{-1}$$

4. Qualitative Graph Reading

From Displacement-Time to Motion

Graph Feature	Meaning
Rising line	positive velocity
Falling line	negative velocity
Flat line	stationary
Increasing steepness	speeding up positive direction

From Velocity-Time to Motion

Graph Feature	Meaning
Horizontal above axis	constant positive velocity
Horizontal below axis	constant negative velocity
Crossing axis	changes direction
Positive slope	positive acceleration

From Acceleration-Time to Motion

Need initial velocity to know exact motion.

Example:

• positive acceleration does not guarantee positive velocity
• object may still move in negative direction while slowing down

5. Graph-to-Motion Translation

Example A

Velocity-time graph is horizontal at $+8$.

Interpretation:

• moving in positive direction
• constant speed
• acceleration = 0

Example B

Velocity-time graph slopes downward from +10 to 0.

Interpretation:

• moving forward
• slowing down uniformly
• constant negative acceleration

Example C

Displacement-time graph has turning point.

Interpretation:

• gradient = 0 there
• instantaneous velocity = 0

6. Units Check (Useful in Exams)

Gradient Units

s-t graph

$$\frac{\text{m}}{\text{s}}={\text{m s}}^{-1}$$

→ velocity

v-t graph

$$\frac{{\text{m s}}^{-1}}{\text{s}}={\text{m s}}^{-2}$$

→ acceleration

Area Units

v-t graph

$$({\text{m s}}^{-1})(\text{s})=\text{m}$$

→ displacement

a-t graph

$$({\text{m s}}^{-2})(\text{s})={\text{m s}}^{-1}$$

→ change in velocity

7. Worked Examples

Example 1

A velocity-time graph rises linearly from 0 to 20 m s${}^{-1}$ in 5 s.

Acceleration

$$a=\frac{20-0}{5}=4.0{\text{m s}}^{-2}$$

Displacement

Area of triangle:

$$\frac{1}{2}(5)(20)=50\text{m}$$

Example 2

A displacement-time graph is horizontal from 3 s to 7 s.

Gradient = 0.

Therefore object is stationary during that interval.

8. Common Exam Pitfalls

• using graph height instead of gradient
• forgetting area below axis is negative displacement
• assuming curved graph means constant acceleration
• confusing displacement with distance
• forgetting tangent needed for instantaneous gradient
• assuming zero velocity means zero acceleration

9. Quick Summary

$$v=\frac{ds}{dt}$$ $$a=\frac{dv}{dt}$$ $$\Delta s=\int vdt$$ $$\Delta v=\int adt$$

• Gradient of s-t graph = velocity
• Gradient of v-t graph = acceleration
• Area under v-t graph = displacement
• Area under a-t graph = change in velocity

Related Links

Links

• Kinematics
• Kinematic Quantities and Sign Conventions
• Constant Acceleration Models
• Projectile and Relative Motion
• Projectile Motion