Speed, Distance and Time Graphs
Graphs are a powerful way to represent motion visually. A distance-time graph shows how far an object has travelled over time, while a speed-time graph shows how the speed changes over time.
By reading these graphs, you can calculate speed, distance, acceleration, and identify periods of rest or uniform motion — all without equations.
This topic covers both types of graphs, how to interpret them, and how to extract numerical information from them.
What is Speed, Distance and Time Graphs?
Key relationships:
Speed = Distance / Time
- Distance-Time Graph: Time on x-axis, distance on y-axis. The slope of the graph gives the speed.
- Speed-Time Graph: Time on x-axis, speed on y-axis. The area under the graph gives the distance travelled.
Speed, Distance and Time Graphs Formula
Reading Distance-Time Graphs:
- Straight line with positive slope → uniform speed. Slope = speed.
- Horizontal line → object is at rest (speed = 0).
- Steeper line → higher speed.
- Curved line → changing speed (acceleration or deceleration).
Reading Speed-Time Graphs:
- Horizontal line → constant speed (uniform motion).
- Line going up → acceleration (speed increasing).
- Line going down → deceleration (speed decreasing).
- Area under the graph = distance covered.
Types and Properties
Types of motion graphs:
- Uniform motion: Distance-time graph is a straight line through the origin. Speed-time graph is a horizontal line.
- Object at rest: Distance-time graph is a horizontal line. Speed-time graph shows zero on y-axis.
- Non-uniform motion: Distance-time graph is curved. Speed-time graph is a non-horizontal line.
- Journey with stops: Distance-time graph has flat portions (horizontal segments) where the object rests.
- Return journey: Distance-time graph goes up (outward) then comes down (return).
Solved Examples
Example 1: Example 1: Reading speed from a distance-time graph
Problem: A distance-time graph shows a straight line from (0, 0) to (4, 80). Find the speed.
Solution:
- Distance = 80 km, Time = 4 hours
- Speed = Distance/Time = 80/4 = 20 km/h
Alternatively: Slope = (80 − 0)/(4 − 0) = 20 km/h.
Answer: Speed = 20 km/h.
Example 2: Example 2: Identifying rest period
Problem: A distance-time graph shows: (0,0) to (2,60) — straight line, (2,60) to (3,60) — horizontal, (3,60) to (5,120) — straight line. Describe the motion.
Solution:
- 0 to 2 hours: Uniform speed = 60/2 = 30 km/h.
- 2 to 3 hours: Distance unchanged → object is at rest for 1 hour.
- 3 to 5 hours: Uniform speed = (120−60)/(5−3) = 60/2 = 30 km/h.
Answer: The object moves at 30 km/h, rests for 1 hour, then moves again at 30 km/h.
Example 3: Example 3: Comparing speeds
Problem: Two objects A and B start from the same point. A reaches 100 km in 5 hours (straight line), B reaches 100 km in 4 hours (straight line). Which is faster?
Solution:
- Speed of A = 100/5 = 20 km/h
- Speed of B = 100/4 = 25 km/h
- B's line is steeper (greater slope = greater speed).
Answer: B is faster at 25 km/h compared to A's 20 km/h.
Example 4: Example 4: Distance from speed-time graph
Problem: A speed-time graph shows a horizontal line at 60 km/h for 3 hours. Find the distance.
Solution:
- Area under the graph = base × height = 3 × 60 = 180
Answer: Distance = 180 km.
Example 5: Example 5: Triangular speed-time graph
Problem: A car accelerates uniformly from 0 to 60 km/h in 2 hours, then decelerates to 0 in the next 1 hour. Find the total distance.
Solution:
The speed-time graph is a triangle with base 3 hours and height 60 km/h (peak at t = 2).
- Area under graph = area of triangle with vertices (0,0), (2,60), (3,0)
- This is a triangle with base = 3 and height = 60, but it is NOT a simple triangle. Split into two parts:
- Part 1 (0 to 2 hours): Triangle with base 2, height 60. Area = ½ × 2 × 60 = 60 km.
- Part 2 (2 to 3 hours): Triangle with base 1, height 60. Area = ½ × 1 × 60 = 30 km.
- Total distance = 60 + 30 = 90 km.
Answer: Total distance = 90 km.
Example 6: Example 6: Average speed from graph
Problem: A person walks 6 km in 2 hours, rests for 1 hour, then walks 9 km in 3 hours. Find the average speed.
Solution:
- Total distance = 6 + 9 = 15 km
- Total time = 2 + 1 + 3 = 6 hours
- Average speed = 15/6 = 2.5 km/h
Answer: Average speed = 2.5 km/h.
Example 7: Example 7: Return journey
Problem: A boy cycles from home (0 km) to school (8 km) in 30 minutes, stays for 4 hours, then returns home in 40 minutes. Sketch the distance-time graph and find speeds.
Solution:
- Going: 8 km in 0.5 h → speed = 16 km/h. Graph: line from (0,0) to (0.5, 8).
- At school: 4 hours. Graph: horizontal line from (0.5, 8) to (4.5, 8).
- Return: 8 km in 2/3 h → speed = 12 km/h. Graph: line from (4.5, 8) to (5⅙, 0).
Answer: Speed going = 16 km/h, speed returning = 12 km/h.
Example 8: Example 8: Linear graph equation
Problem: The distance-time relationship for a car is d = 40t, where d is in km and t in hours. Draw the graph and find the distance after 3.5 hours.
Solution:
- This is a straight line through the origin with slope 40.
- At t = 1: d = 40. At t = 2: d = 80. At t = 3.5: d = 140.
Answer: Distance after 3.5 hours = 140 km.
Example 9: Example 9: Speed from a curved graph
Problem: A distance-time graph is curved (concave upward). What does this tell us about the motion?
Solution:
- A concave-upward curve means the slope is increasing over time.
- Since slope = speed, the speed is increasing.
- This indicates acceleration (the object is speeding up).
Answer: The object is accelerating.
Example 10: Example 10: Trapezoidal speed-time graph
Problem: A car accelerates from 0 to 40 km/h in 1 hour, maintains 40 km/h for 2 hours, then decelerates to 0 in 1 hour. Find total distance.
Solution:
The speed-time graph is a trapezoid.
- Area of acceleration triangle = ½ × 1 × 40 = 20 km
- Area of constant speed rectangle = 2 × 40 = 80 km
- Area of deceleration triangle = ½ × 1 × 40 = 20 km
- Total = 20 + 80 + 20 = 120 km
Answer: Total distance = 120 km.
Real-World Applications
Real-life applications:
- Traffic analysis: Speed cameras and GPS data produce speed-time profiles of vehicles.
- Sports: Analysing an athlete's speed during a race using timing data.
- Aviation: Flight data recorders plot altitude-time and speed-time graphs.
- Train schedules: Distance-time graphs show expected arrival times at each station.
- Physics experiments: Motion sensors produce real-time distance-time graphs.
Key Points to Remember
- In a distance-time graph, slope = speed.
- In a speed-time graph, area under the curve = distance.
- A horizontal line on a distance-time graph means the object is at rest.
- A steeper line means greater speed.
- A curved line on a distance-time graph means non-uniform speed.
- Speed = Distance / Time. Use consistent units.
- Average speed = Total distance / Total time (including rest periods).
- The speed-time graph of uniform acceleration is a straight line with positive slope.
Practice Problems
- A car travels 150 km in 3 hours at constant speed. Draw the distance-time graph and find the slope.
- From a distance-time graph, the coordinates are (1, 20) and (4, 80). Find the speed.
- A speed-time graph shows speed = 50 km/h for 4 hours. Find the distance.
- A cyclist rides at 12 km/h for 2 hours, rests for 30 minutes, then rides at 15 km/h for 1 hour. Draw the distance-time graph.
- A car accelerates from 0 to 80 km/h in 2 hours (uniformly). Find the distance using the area of the speed-time triangle.
- From a distance-time graph, what does a downward sloping line indicate?
- Draw speed-time and distance-time graphs for a bus that travels at 40 km/h for 1 hour, stops for 30 min, then travels at 60 km/h for 1.5 hours.
Frequently Asked Questions
Q1. What does the slope of a distance-time graph represent?
The slope represents speed. A steeper slope means higher speed. Zero slope (horizontal) means the object is at rest.
Q2. What does the area under a speed-time graph represent?
The area under the speed-time graph gives the total distance travelled during that time interval.
Q3. What does a horizontal line on a distance-time graph mean?
The object is stationary (at rest). The distance is not changing.
Q4. What does a curved distance-time graph indicate?
Non-uniform speed. If the curve bends upward (concave up), the object is accelerating. If it bends downward (concave down), it is decelerating.
Q5. How do you find average speed from a graph?
Average speed = Total distance / Total time. Read the total distance and total time from the graph.
Q6. Can a distance-time graph go downward?
Yes, if the object is returning to its starting point. The distance from the starting point decreases, so the graph slopes downward.
Q7. What shape is the speed-time graph for uniform acceleration?
A straight line with positive slope, starting from the initial speed.
Q8. What is the difference between speed and velocity?
Speed is scalar (magnitude only). Velocity is vector (magnitude + direction). In Class 8, we mainly use speed.
