Graphs in practical situations
Section: Algebra and graphs 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Distance-Time Graphs
A distance-time graph shows how the distance travelled by a moving object changes over time. The steepness (gradient) of the line represents speed.
- A straight line means constant speed - the steeper the line, the faster the speed
- A horizontal section means the object is stationary (not moving) for that time
- The gradient of a distance-time graph = speed (change in distance ÷ change in time)
- A negative gradient (the line sloping downward) means the object is returning to its starting point
Worked Example: Finding Speed from a Distance-Time Graph
- Question: A cyclist's journey is shown on a distance-time graph. In the first 40 minutes, the cyclist travels 12 km at a constant speed. Find the cyclist's speed in km/h.
- Step 1: Convert the time to hours: 40 min=(40/60)=(2/3) h
- Step 2: Speed = distance ÷ time: 12÷(2/3)=18
- Answer: 18 km/h
Worked Example: Interpreting a Multi-Stage Journey
- Question: A distance-time graph shows a runner's distance from home over 90 minutes. From 0 to 30 minutes, the distance rises steadily from 0 to 6 km. From 30 to 45 minutes, the distance stays at 6 km. From 45 to 90 minutes, the distance falls steadily from 6 km back to 0. Find the runner's speed on the return journey.
- Step 1: From 0 to 30 minutes the runner moves away from home at a constant speed; from 30 to 45 minutes the runner is stationary (resting)
- Step 2: The return journey takes 90-45=45 min=0.75 h
- Step 3: Speed = distance ÷ time: 60.75=8
- Answer: 8 km/h
Common Mistakes
MistakeReading a horizontal section as "no distance travelled during the whole journey" instead of "stationary for a period of time"
Fixa horizontal line means time is passing but distance is not changing - the object has stopped, not failed to move at any point in the journey
MistakeForgetting to convert minutes to hours before dividing distance by time when speed is required in km/h
Fixcheck the units required in the question and convert time (or distance) before calculating
Conversion Graphs
A conversion graph is a straight-line graph used to convert between two related quantities, such as two currencies or two units of measurement.
- To convert a value, find it on one axis, trace across to the line, then read off the corresponding value on the other axis
- The gradient of a conversion graph gives the conversion rate between the two quantities
- A conversion graph that passes through the origin means zero of one quantity corresponds to zero of the other
Worked Example: Using a Conversion Graph
- Question: A conversion graph passes through the origin and shows that 4 miles is equivalent to 6.4 kilometres. Use the graph to convert 10 miles to kilometres.
- Step 1: Find the conversion rate (gradient): 6.44=1.6 km per mile
- Step 2: Multiply by the number of miles: 101.6=16
- Answer: 16 km
Common Mistakes
MistakeReading off the wrong axis, e.g. converting kilometres to miles using the miles-to-kilometres rate instead of its reverse
Fixcheck which axis represents which quantity, and which direction the conversion is required
MistakeAssuming every conversion graph passes through the origin, even for quantities that don't start at zero together (such as Celsius and Fahrenheit)
Fixcheck where the line actually crosses each axis before assuming a simple direct-proportion relationship
Speed-Time Graphs: Gradient and Area
A speed-time graph shows how an object's speed changes over time. Unlike a distance-time graph, the gradient here represents acceleration, and the area underneath the graph represents distance travelled.
- The gradient of a speed-time graph = acceleration (change in speed ÷ change in time)
- A positive gradient means the object is accelerating (speeding up); a negative gradient means it is decelerating (slowing down)
- A horizontal section means constant speed (zero acceleration)
- The area between the graph and the time axis gives the total distance travelled
Worked Example: Finding Acceleration
- Question: A car accelerates from 10 m/s to a speed of V m/s over 25 seconds, at a constant acceleration of 0.8 m/s². Show that V = 30 m/s.
- Step 1: Acceleration = change in speed ÷ time taken: (V-10/25)=0.8
- Step 2: Multiply both sides by 25: V-10=0.825=20
- Step 3: Add 10 to both sides: V=20+10=30
- Answer: V = 30 m/s (shown)
Worked Example: Finding Distance from the Area Under the Graph
- Question: The speed-time graph of a journey rises in a straight line from (0, 0) to (10, 20), stays constant at 20 m/s from 10 to 25 seconds, then the journey ends. Find the total distance travelled during the 25 seconds.
- Step 1: From t = 0 to t = 10: triangle area =(1/2)1020=100 m
- Step 2: From t = 10 to t = 25: rectangle area =1520=300 m
- Step 3: Total distance: 100+300=400
- Answer: 400 m
Common Mistakes
MistakeUsing "area = distance" on a distance-time graph, or "gradient = speed" on a speed-time graph - mixing up the rules for the two graph types
Fixcheck which quantity is on the vertical axis before deciding whether gradient or area is needed
MistakeTreating a trapezium-shaped section of a speed-time graph as a simple rectangle or triangle, and missing part of the area
Fixsplit the graph into simple shapes matching each straight section, and add the areas together
Curved Speed-Time Graphs and Estimating Area
Extended Only
When acceleration is not constant, a speed-time graph curves rather than forming straight lines. The area under a curved section cannot be found using a simple formula, so it is estimated instead.
- A curved section of a speed-time graph shows non-uniform (changing) acceleration
- The gradient at a single point on a curve can be estimated by drawing a tangent to the curve at that point and finding its gradient
- The area under a curved section is estimated by counting the whole and part squares between the curve and the time axis, then multiplying by the value each square represents
Worked Example: Estimating Distance Under a Curve
- Question: A speed-time graph is drawn on a grid where each square represents 3 seconds horizontally and 4 m/s vertically. A curved section of the graph is found to cover approximately 15 whole squares. Estimate the distance travelled during this section.
- Step 1: Each square represents an area of 34=12 m
- Step 2: Total distance ≈ 1512=180
- Answer: approximately 180 m
Common Mistakes
MistakeCounting only whole squares and ignoring partial squares along the curve, which noticeably underestimates the area
Fixcount part-squares too, combining pairs of small pieces to estimate whole squares where possible
MistakeMultiplying the number of squares by the number of seconds only, forgetting each square represents an area (time × speed)
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