Chapter 7: Graphs of Motion

Overview

Welcome to Chapter 7 of Form 4 Mathematics! This chapter introduces you to the fascinating world of motion graphs. You'll learn to interpret and analyze distance-time and speed-time graphs, understand the relationship between them, and solve real-world motion problems. These skills are essential for physics, engineering, and understanding how objects move.

What You'll Learn:

Interpret and sketch distance-time graphs
Interpret and sketch speed-time graphs
Relate areas under graphs to distance traveled
Solve motion problems using graphical analysis

Learning Objectives

After completing this chapter, you will be able to:

Sketch, interpret, and solve problems involving distance-time graphs
Sketch, interpret, and solve problems involving speed-time graphs, including making connections between area under graphs and distance traveled

Key Concepts

Distance-Time Graphs

Distance-time graphs show how distance from a starting point changes over time.

Mathematical Representation:

Y-axis: Distance (s)
X-axis: Time (t)
Relationship: $s = f(t)$

Axes:

Y-axis: Distance
X-axis: Time

Key Features:

Graph Type	Meaning	Mathematical Interpretation
Positive slope	Moving away from starting point	$\frac{\Delta s}{\Delta t} > 0$
Negative slope	Moving toward starting point	$\frac{\Delta s}{\Delta t} < 0$
Zero slope (horizontal)	Stationary (not moving)	$\frac{\Delta s}{\Delta t} = 0$
Steep slope	High speed	Large $\lvert\frac{\Delta s}{\Delta t}\rvert$
Gentle slope	Low speed	Small $\lvert\frac{\Delta s}{\Delta t}\rvert$

Gradient (Slope): The gradient represents speed:

Gradient = $\frac{\text{Change in distance}}{\text{Change in time}} = \frac{\Delta s}{\Delta t} = v$ (velocity/speed)

Visual Examples:

Speed-Time Graphs

Speed-time graphs show how speed changes over time.

Mathematical Representation:

Y-axis: Speed (v)
X-axis: Time (t)
Relationship: $v = f(t)$
Area under graph represents: Distance = $\int v \, dt$

Axes:

Y-axis: Speed
X-axis: Time

Key Features:

Graph Type	Meaning	Mathematical Interpretation
Positive slope	Acceleration	$\frac{\Delta v}{\Delta t} > 0$
Negative slope	Deceleration	$\frac{\Delta v}{\Delta t} < 0$
Zero slope (horizontal)	Constant speed	$\frac{\Delta v}{\Delta t} = 0$
Increasing area	Distance accumulating	Area grows over time
Constant area growth	Constant speed	Linear area increase

Gradient (Slope): The gradient represents acceleration:

Gradient = $\frac{\text{Change in speed}}{\text{Change in time}} = \frac{\Delta v}{\Delta t} = a$ (acceleration)

Area Under Graph: The area under a speed-time graph represents distance traveled:

Area = $\text{base} \times \text{height}$ (rectangle)
Area = $\frac{1}{2} \times \text{base} \times \text{height}$ (triangle)
Area = $\frac{1}{2} \times (a + b) \times \text{height}$ (trapezoid)

Area Under Speed-Time Graphs

The area under a speed-time graph represents the distance traveled during that time period.

Mathematical Principle:

\text{Distance} = \int_{t_1}^{t_2} v(t) \, dt = \text{Area under curve from } t_1 \text{ to } t_2

Common Shapes and Areas:

Shape	Distance Formula	Mathematical Expression	Physical Interpretation
Rectangle	Area = base × height	$d = v \times t$	Constant speed motion
Triangle	Area = ½ × base × height	$d = \frac{1}{2} \times v \times t$	Uniform acceleration from rest
Trapezoid	Area = ½ × (a + b) × h	$d = \frac{1}{2} \times (v_1 + v_2) \times t$	Changing speed over time

Visual Area Analysis:

Important Formulas and Methods

Distance-Time Graph Calculations

Gradient (Speed):

\text{Speed} = \frac{\text{Change in distance}}{\text{Change in time}} = \frac{y_2 - y_1}{x_2 - x_1}

Average Speed:

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

Speed-Time Graph Calculations

Gradient (Acceleration):

\text{Acceleration} = \frac{\text{Change in speed}}{\text{Change in time}} = \frac{y_2 - y_1}{x_2 - x_1}

Distance from Area:

Rectangle: Distance = speed × time
Triangle: Distance = ½ × base × height
Trapezoid: Distance = ½ × (sum of parallel sides) × height

Key Relationships:

Positive gradient = acceleration
Negative gradient = deceleration
Zero gradient = constant speed

Motion Equations

The graphical analysis corresponds to kinematic equations:

Distance = Speed × Time
Acceleration = (Final Speed - Initial Speed) / Time
Distance under constant acceleration: $s = ut + \frac{1}{2}at^2$

Step-by-Step Solved Examples

Example 1: Distance-Time Graph Analysis

Problem: A car travels according to the distance-time graph:

From t=0 to t=2: Distance from 0 to 60 km
From t=2 to t=4: Distance remains 60 km (stationary)
From t=4 to t=6: Distance from 60 to 120 km
From t=6 to t=8: Distance from 120 to 80 km (returning)

Find: a) Speed during each time interval b) Average speed for the entire journey

Solution: a) Speed during each interval:

0-2 hours: Speed = (60 - 0)/(2 - 0) = 30 km/h
2-4 hours: Speed = (60 - 60)/(4 - 2) = 0 km/h (stationary)
4-6 hours: Speed = (120 - 60)/(6 - 4) = 30 km/h
6-8 hours: Speed = (80 - 120)/(8 - 6) = -20 km/h (returning)

b) Average speed: Total distance = 80 km (final position) Total time = 8 hours Average speed = 80 / 8 = 10 km/h

Answer: Average speed is 10 km/h

Example 2: Speed-Time Graph Analysis

Problem: A train's speed-time graph shows:

From t=0 to t=3: Speed increases from 0 to 30 m/s (acceleration)
From t=3 to t=7: Speed constant at 30 m/s
From t=7 to t=10: Speed decreases from 30 to 0 m/s (deceleration)

Find the total distance traveled.

Solution: Distance = Area under graph:

Triangle (0-3s): Area = ½ × 3 × 30 = 45 m
Rectangle (3-7s): Area = 4 × 30 = 120 m
Triangle (7-10s): Area = ½ × 3 × 30 = 45 m

Total distance: 45 + 120 + 45 = 210 m

Answer: Total distance traveled is 210 meters

Example 3: Complex Motion Problem

Problem: A ball is thrown upwards with initial speed of 20 m/s. The speed-time graph shows:

Initial speed: 20 m/s (upwards)
Deceleration due to gravity: 10 m/ $s^2$ (downwards)
Reaches maximum height when speed = 0
Falls back down with increasing speed

Find: a) Time to reach maximum height b) Maximum height reached c) Total distance when it returns to starting point

Solution: a) Time to reach maximum height: Using v = u + at, where v = 0, u = 20, a = -10 0 = 20 + (-10)t → 10t = 20 → t = 2 seconds

b) Maximum height: Using $s = ut + \frac{1}{2}at^2$ $s = 20(2) + \frac{1}{2}(-10)(2)^2 = 40 - 20 = 20 \text{ meters}$

c) Total distance when returning: The ball travels up 20m and down 20m, so total distance = 40m

Answer: a) 2 seconds, b) 20 meters, c) 40 meters

Example 4: Real-world Application

Problem: A delivery van makes a delivery route:

Accelerates from rest at 2 m/ $s^2$ for 10 seconds
Travels at constant speed for 30 seconds
Decelerates at 3 m/ $s^2$ until stopped

Find: a) Maximum speed reached b) Total distance traveled c) Time taken to stop

Solution: a) Maximum speed: v = u + at = 0 + 2(10) = 20 m/s

b) Total distance:

Acceleration phase: $s = ut + \frac{1}{2}at^2 = 0 + \frac{1}{2}(2)(10)^2 = 100 \text{ m}$
Constant speed: $s = vt = 20 \times 30 = 600 \text{ m}$
Deceleration phase:
- $v = 0, u = 20, a = -3$
- $0 = 20 + (-3)t \rightarrow t = 20/3 \approx 6.67 \text{ s}$
- $s = ut + \frac{1}{2}at^2 = 20(6.67) + \frac{1}{2}(-3)(6.67)^2 \approx 133.3 - 66.7 = 66.6 \text{ m}$

Total distance: 100 + 600 + 66.6 = 766.6 m

Answer: a) 20 m/s, b) 766.6 m, c) 6.67 s

Example 5: Interpreting Complex Graphs

Problem: The speed-time graph below shows a car's journey:

Speed (m/s)
40 |     /\
30 |    /  \
20 |   /    \
10 |  /      \
 0 |/________\_______
   0  5  10  15  20 Time (s)

Describe the motion and find total distance.

Solution: Motion description:

0-5s: Uniform acceleration from 0 to 20 m/s
5-10s: Constant speed of 20 m/s
10-15s: Uniform deceleration from 20 to 10 m/s
15-20s: Constant speed of 10 m/s

Distance calculation:

Triangle (0-5s): Area = ½ × 5 × 20 = 50 m
Rectangle (5-10s): Area = 5 × 20 = 100 m
Trapezoid (10-15s): Area = ½ × (20 + 10) × 5 = 75 m
Rectangle (15-20s): Area = 5 × 10 = 50 m

Total distance: 50 + 100 + 75 + 50 = 275 m

Answer: Total distance is 275 meters

Real-world Applications

1. Transportation Applications

Applications:

Traffic Analysis: Speed limits and traffic flow optimization
Vehicle Design: Acceleration and braking performance curves
Route Planning: Time and distance optimization using GPS data
Public Transit: Efficient scheduling and speed management

2. Sports Performance Analysis

Applications:

Athlete Performance: Sprint times, running speeds, acceleration rates
Equipment Design: Ball trajectories, racket speeds, athletic footwear
Training Analysis: Speed and endurance improvements over time
Game Strategy: Pacing and timing in various sports

3. Engineering Applications

Applications:

Machine Design: Moving parts and mechanisms with precise motion control
Robotics: Robot arm movements and speed optimization
Transportation Systems: Elevators, conveyor belts, automated production lines
Manufacturing: Production rate optimization and efficiency analysis

4. Physics and Scientific Research

Applications:

Projectile Motion: Ball throwing, rocket trajectories, optimal launch angles
Wave Analysis: Wave speed, frequency, and propagation studies
Particle Physics: Particle motion and collision dynamics
Astronomy: Orbital mechanics and celestial body motion analysis

5. Everyday Life Applications

Applications:

Transportation: Car fuel efficiency, journey planning, speed tracking
Exercise: Running pace, cycling speed, heart rate monitoring
Weather: Wind speed, rainfall rates, temperature changes
Technology: Internet speeds, data transfer rates, device performance metrics

Important Terms

Term	Definition	Example
Distance-Time Graph	Shows distance from start over time	Car's journey from home
Speed-Time Graph	Shows speed over time	Train's acceleration profile
Gradient	Slope of the graph	Speed for distance-time, acceleration for speed-time
Acceleration	Rate of change of speed	Car speeding up from 0 to 60 km/h
Deceleration	Negative acceleration (slowing down)	Car braking to stop
Area Under Graph	Distance traveled in speed-time graphs	Rectangle area = constant speed × time
Constant Speed	Horizontal line in distance-time graph	Cruise control on highway
Stationary	Horizontal line in distance-time graph	Car stopped at traffic light

Summary Points

Distance-Time Graph:
- Y-axis: Distance, X-axis: Time
- Gradient = Speed
- Horizontal line = Stationary
Speed-Time Graph:
- Y-axis: Speed, X-axis: Time
- Gradient = Acceleration
- Horizontal line = Constant speed
- Area = Distance traveled
Key Relationships:
- Positive gradient = acceleration/speed increase
- Negative gradient = deceleration/speed decrease
- Zero gradient = constant speed/stationary
Area Calculations:
- Rectangle: base × height
- Triangle: ½ × base × height
- Trapezoid: ½ × (a + b) × height

Practice Tips for SPM Students

1. Graph Interpretation

Practice reading information from graphs
Learn to identify key features (slopes, areas)
Understand what different graph shapes represent

2. Calculation Methods

Master gradient calculations
Practice area calculations for different shapes
Learn to convert between different motion parameters

3. Real-world Scenarios

Practice relating graphs to real situations
Understand everyday motion problems
Learn to interpret motion data

4. Common Mistakes to Avoid

Confusing distance-time with speed-time graphs
Misinterpreting gradient meanings
Forgetting that area under speed-time = distance
Using wrong units in calculations

SPM Exam Tips

Paper 1 (Multiple Choice)

Look for key graph features
Remember gradient meanings for different graph types
Practice quick area calculations
Understand motion terminology

Paper 2 (Structured)

Show all gradient and area calculations
Label graph axes clearly
Explain motion phases in context
Use units consistently throughout

Did You Know? The study of motion dates back to ancient Greece with Aristotle, but it was Galileo Galilei in the 17th century who first used mathematical analysis to study motion systematically. His work laid the foundation for Newton's laws of motion and modern physics!

Next Chapter: In Chapter 8, you'll explore measures of dispersion for ungrouped data and learn to analyze data spread using range, quartiles, variance, and standard deviation.