How to Solve: Real-Life Graphs (Distance-Time, Velocity-Time, Conversions)

How to Solve: Real-Life Graphs (Distance-Time, Velocity-Time, Conversions)

For GCSE & A-Level Maths (Edexcel/AQA/OCR)

Introduction

"Mastering distance-time and velocity-time graphs could earn you 10-15% of your GCSE Maths paper—and they’re the key to real-world problems like calculating journey times, fuel efficiency, or even Olympic sprint speeds. One wrong gradient or misread axis, and you lose easy marks. Let’s fix that."

What You Need To Know First

1. Gradient (slope) – How steep a line is; calculated as rise ÷ run.
2. Units – Distance (m/km), time (s/h), speed (m/s or km/h).
3. Basic algebra – Rearranging equations like speed = distance ÷ time.

Key Vocabulary

Formulas To Know

Step-by-Step Method

For Distance-Time Graphs

1. Label axes: x-axis = time, y-axis = distance.
2. Identify sections: Flat line = stationary. Straight line = constant speed.
3. Calculate speed: Gradient = change in distance ÷ change in time.
4. Check units: Convert if needed (e.g., m/s → km/h).

For Velocity-Time Graphs

1. Label axes: x-axis = time, y-axis = velocity.
2. Identify sections: Flat line = constant speed. Sloped line = acceleration/deceleration.
3. Calculate acceleration: Gradient = change in velocity ÷ change in time.
4. Find distance: Area under graph (split into rectangles/triangles/trapeziums).
5. Check direction: Negative velocity = moving backward.

For Unit Conversions

1. Write the conversion factor: e.g., 1 km = 1000 m, 1 h = 3600 s.
2. Multiply/divide: To convert km/h → m/s, ÷ 3.6. To convert m/s → km/h, × 3.6.
3. Check units match: Always write units in your final answer.

Worked Examples

Example 1 – Basic (Distance-Time Graph)

Question: A car travels 60 km in 1.5 hours, stops for 30 minutes, then drives 40 km in 1 hour. Sketch the distance-time graph and find the average speed for the whole journey.

Steps: 1. Plot points:
- (0 h, 0 km) → (1.5 h, 60 km) → (2 h, 60 km) → (3 h, 100 km). 2. Calculate speeds:
- First section: 60 km ÷ 1.5 h = 40 km/h.
- Second section: 0 km ÷ 0.5 h = 0 km/h (stationary).
- Third section: 40 km ÷ 1 h = 40 km/h. 3. Average speed:
- Total distance = 60 + 0 + 40 = 100 km.
- Total time = 3 h.
- Average speed = 100 km ÷ 3 h ≈ 33.3 km/h.

What we did and why: - Broke the journey into sections to find speeds. - Used total distance ÷ total time for average speed (not average of speeds).

Example 2 – Medium (Velocity-Time Graph)

Question: A cyclist accelerates from rest to 12 m/s in 6 s, then brakes to 4 m/s in 4 s. Sketch the velocity-time graph and find: a) The acceleration in the first 6 s. b) The total distance travelled.

Steps: 1. Plot points:
- (0 s, 0 m/s) → (6 s, 12 m/s) → (10 s, 4 m/s). 2. a) Acceleration (first 6 s):
- Gradient = (12 – 0) ÷ (6 – 0) = 2 m/s². 3. b) Total distance:
- Area under graph = triangle (0-6 s) + trapezium (6-10 s).
- Triangle: 0.5 × 6 × 12 = 36 m.
- Trapezium: 0.5 × (12 + 4) × 4 = 32 m.
- Total distance = 36 + 32 = 68 m.

What we did and why: - Used gradient for acceleration (rise ÷ run). - Split area into shapes to calculate distance (velocity × time).

Example 3 – Exam-Style (Unit Conversion + Graph)

Question: A train travels at 20 m/s for 5 minutes, then decelerates at 0.5 m/s² for 20 s. Find: a) The distance covered in the first 5 minutes (in km). b) The final speed after deceleration.

Steps: 1. a) Distance in first 5 minutes:
- Convert time: 5 minutes = 300 s.
- Distance = speed × time = 20 m/s × 300 s = 6000 m.
- Convert to km: 6000 m ÷ 1000 = 6 km. 2. b) Final speed after deceleration:
- Initial speed = 20 m/s.
- Deceleration = 0.5 m/s² for 20 s.
- Change in speed = 0.5 × 20 = 10 m/s.
- Final speed = 20 – 10 = 10 m/s.

What we did and why: - Converted units first (minutes → seconds, m → km). - Used acceleration × time to find change in speed.

Common Mistakes

Exam Traps

1-Minute Recap

"Right, listen up—this is your 60-second cheat sheet for real-life graphs. For distance-time graphs, gradient = speed. Flat line? Stationary. For velocity-time graphs, gradient = acceleration, and area = distance. Always check units—km/h to m/s? Divide by 3.6. Negative velocity? Moving backward. Examiners love hiding stationary periods or non-linear scales, so label everything. And remember: average speed is total distance ÷ total time, not the average of speeds. Got it? Good. Now go smash those graphs."