🏠 Home › 📚 CSEC Mathematics › 📊 Graphs of Motion

Distance-Time and Speed-Time Graphs

The Language of Motion

Essential Understanding: Graphs tell the story of how objects move! By learning to "read" distance-time and speed-time graphs, you can understand everything from a car journey to a sprinter's race—all without being there!

🎯 Key Variable Understanding

Time (t): Always the independent variable — it goes on the x-axis (horizontal). Time always increases from left to right.
Distance (d): The dependent variable on distance-time graphs — goes on the y-axis (vertical). Shows how far you've traveled from the starting point.
Speed (v): The dependent variable on speed-time graphs — goes on the y-axis. Shows how fast you're moving at each moment.

\[ \text{Key Concept: The slope (gradient) represents a "rate of change"} \]

📏 Distance-Time Graph

Slope = Speed

\[ m = \frac{\text{rise}}{\text{run}} = \frac{\Delta d}{\Delta t} = \text{Speed} \]

A steeper slope means faster speed!

🚀 Speed-Time Graph

Slope = Acceleration

\[ m = \frac{\text{rise}}{\text{run}} = \frac{\Delta v}{\Delta t} = \text{Acceleration} \]

A negative slope means deceleration!

Distance-Time Graphs: The Basics

A distance-time graph tells the complete story of a journey. The line shows exactly what happened at every moment. Learn to read these three key patterns:

⏸️

Horizontal Line: Stationary

What it means: The object is not moving. Distance stays the same while time passes.

Real-world example: A car stopped at traffic lights, a student sitting in class, a plane on the runway before takeoff.

Slope: 0 (flat, horizontal line)

➡️

Straight Sloped Line: Constant Speed

What it means: The object is moving at the same speed throughout. Neither speeding up nor slowing down.

Real-world example: A car on a highway cruise control, a train between stations, a person walking at a steady pace.

Slope: Constant positive value

📈

Curved Line: Changing Speed

What it means: The object is accelerating (curving upward) or decelerating (curving toward horizontal).

Real-world example: A car accelerating from rest, a runner finishing a race, a bicycle going downhill.

Slope: Changing constantly!

The Speed Formula

Speed is calculated the same way you find the gradient of a line!

                \[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]
            

This is identical to the gradient formula you learned for linear graphs!

Find Two Points: Choose two clear points on the line where coordinates are easy to read.

Calculate the Rise: Subtract the first y-value from the second y-value (Distance₂ − Distance₁).

Calculate the Run: Subtract the first x-value from the second x-value (Time₂ − Time₁).

Divide Rise by Run: Speed = Rise ÷ Run. Don't forget your units (usually m/s or km/h)!

Speed-Time Graphs: Advanced Analysis

Speed-time graphs give us even more power—they can tell us both acceleration AND the total distance traveled!

⚡

Gradient = Acceleration

Positive Gradient: Object is speeding up (accelerating).

Zero Gradient: Object is moving at constant speed.

Negative Gradient: Object is slowing down (decelerating).

Formula: \[ a = \frac{\Delta v}{\Delta t} \]

📐

Area Under the Curve = Distance

The Key Insight: The total distance traveled is equal to the area between the graph line and the x-axis (time axis).

Why? Area = Base × Height = Time × Speed = Distance!

This is a CORE CSEC SKILL you must master.

🔺

Geometric Breakdown

Rectangle: Constant speed journey. Area = base × height.

Triangle: Accelerating from rest. Area = ½ × base × height.

Trapezium: Speed changing from u to v. Area = ½ × (sum of parallel sides) × height.

Area Formulas Reference

Rectangle \[ \text{Area} = b \times h = \text{Time} \times \text{Speed} \]

For constant speed journeys

Triangle \[ \text{Area} = \frac{1}{2} \times b \times h \]

For acceleration from rest

Trapezium \[ \text{Area} = \frac{1}{2} \times (a+b) \times h \]

For changing speed (a and b are speeds)

Interactive "Journey Lab"

🔬

Explore Motion: Speed-Time Graph Simulator

Objective: Adjust the speed parameters to see how the graph changes. Watch how the area under the curve represents total distance traveled!

Journey Controls

Initial Speed: 0 m/s

0 m/s 10 m/s 20 m/s

Current Speed (m/s)

Acceleration (m/s²)

Total Distance (m)

Total Time (s)

Compare Mode: Distance-Time vs Speed-Time

📊

Side-by-Side Graph Comparison

Understanding the Connection: See how the same journey looks different on Distance-Time and Speed-Time graphs. Watch how changes in speed affect the slope of the distance graph!

Distance (m)

Speed (m/s)

Time (s)

CSEC Exam Mastery Arena

Test Your Understanding

A car travels 100m in 20 seconds. What is its average speed?

0.2 m/s

5 m/s

2000 m/s

50 m/s

Solution: Speed = Distance ÷ Time = 100 ÷ 20 = 5 m/s. Remember: always divide distance by time, not the other way around!

A speed-time graph shows a triangle with base 8s and height 20m/s. What is the total distance traveled?

40 m

160 m

80 m

28 m

Solution: Area of triangle = ½ × base × height = ½ × 8 × 20 = 80m. This represents an object accelerating from rest (0 m/s) to 20 m/s.

A journey has three parts: 100m at 10 m/s, then 200m at 15 m/s, then 50m at 5 m/s. What is the average speed for the whole trip?

10 m/s (average of 10, 15, 5)

9.375 m/s

12.5 m/s

15 m/s

Solution: Average Speed = Total Distance ÷ Total Time.
Total Distance = 100 + 200 + 50 = 350m
Total Time = (100÷10) + (200÷15) + (50÷5) = 10 + 13.33 + 10 = 33.33s
Average Speed = 350 ÷ 33.33 = 10.5 m/s (actually 10.5 m/s based on exact calculation: 350 ÷ 33.333... = 10.5)
⚠️ Common Mistake: Never just average the individual speeds! (10+15+5)÷3 = 10 m/s is WRONG!

What does a horizontal line on a speed-time graph represent?

The object is stopped

Constant speed (not accelerating)

Maximum speed reached

The object is returning to start

Explanation: A horizontal line on a speed-time graph means speed is NOT changing. This is constant speed, NOT being stationary! (Stationary would be a horizontal line ON the x-axis at speed = 0).

🎯

CSEC Examination Mastery Tip: Drawing Tangents

For Non-Linear Motion: When you need to find instantaneous speed from a curved distance-time graph, you must draw a tangent:

Place your ruler at the point where you want the speed
Rotate the ruler until it just touches the curve at that ONE point
Draw the tangent line (make it long enough to easily read coordinates)
Calculate the gradient of this tangent line

Pro Tip: Your tangent should touch the curve at exactly one point and follow the curve's direction at that moment—like a snapshot of the instantaneous motion!

📝 Worked Example: Multi-Stage Journey

Question: A car travels 60km in 1 hour, then 40km in 30 minutes. Calculate the average speed for the whole journey.

Calculate Total Distance: 60km + 40km = 100km

Convert Units if Needed: Time 1 = 1 hour, Time 2 = 30 minutes = 0.5 hours

Calculate Total Time: 1 + 0.5 = 1.5 hours

Calculate Average Speed: Total Distance ÷ Total Time = 100km ÷ 1.5 hours = 66.7 km/h

⚠️ Common Pitfalls to Avoid

Don't lose precious marks! Watch out for these common mistakes:

❌ Confusing "Stationary" on Different Graphs

The Mistake: Thinking a horizontal line always means "stopped."

✅ The Fix: On distance-time: horizontal = stopped. On speed-time: horizontal = constant speed (could be fast!). Only at y=0 is the object truly stationary.

❌ Forgetting Unit Conversions

The Mistake: Mixing minutes with hours, or meters with kilometers.

✅ The Fix: ALWAYS check units before calculating. 30 minutes = 0.5 hours, 2 km = 2000m. Write conversions clearly in your working.

❌ Averaging Speeds Instead of Distances

The Mistake: Using (speed₁ + speed₂) ÷ 2 instead of Total Distance ÷ Total Time.

✅ The Fix: Average Speed = Total Distance ÷ Total Time. This is NEVER simply the average of individual speeds unless times are equal!

❌ Mixing Up Slope and Area

The Mistake: Using area for distance-time or slope for speed-time distance calculations.

✅ The Fix: Distance-Time: Use SLOPE for speed. Speed-Time: Use AREA for distance. MEMORIZE THIS!

📝 Chapter Summary

Quick Reference: Distance-Time Graph

Horizontal line	Stationary (not moving)
Straight sloping line	Constant speed
Curved line (upward)	Accelerating (speeding up)
Slope	Speed = rise ÷ run

Quick Reference: Speed-Time Graph

Horizontal on x-axis	Stationary (speed = 0)
Horizontal above x-axis	Constant speed
Positive slope	Acceleration
Negative slope	Deceleration
Area under curve	Total distance traveled