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E2.9 Graphs in practical situations

Syllabus
0580–2028–2029
Topic
E2.9
Level
Extended

Learning objectives

E2.9.1Use and interpret graphs in practical situations including travel• Use and interpret graphs in practical situations including travel graphs and conversion graphs.E2.9.2Draw graphs from given data.• Draw graphs from given data.E2.9.3Apply the idea of rate of change to simple kinematics involving• Apply the idea of rate of change to simple kinematics involving distance–time and speed–time graphs, acceleration and deceleration.E2.9.4Calculate distance travelled as area under a speed–time graph• Calculate distance travelled as area under a speed–time graph. Includes estimation and interpretation of the gradient of a tangent at a point. Areas will involve linear sections of the graph only.

Interpret travel and conversion graphs

A practical graph connects two measured quantities. Read the axes, units and scale first; then interpret a coordinate, interval or gradient in the context rather than as an isolated number.

Distance–time feature Meaning
rising straight segment moving away at constant speed
falling straight segment moving back at constant speed
horizontal segment stationary
steeper segment greater speed
intersection of two journeys same place at the same time

To read a value, start at the known coordinate, move parallel to an axis until reaching the graph, then move parallel to the other axis and read the scale. Interpolate carefully between labelled marks.

A conversion graph maps one unit or currency to another. Read in either direction using the same line; a straight line through the origin represents a constant conversion factor.

A downward distance–time segment means returning toward the reference point, not travelling at negative speed. A horizontal segment means stopped, not zero distance from the start.

Draw a practical graph from data

A graph should preserve every supplied value and make the relationship readable. For a journey, each segment must begin where the preceding event ends.

Label both axes with quantity and unit; choose a uniform scale covering all data; plot each coordinate accurately; join points with straight segments when the rate is constant or as directed; check endpoints, stops and continuity against the context.

Journey statement Graph action
starts later first point has the stated later time
travels at constant speed draw a straight sloping segment
rests for a time interval draw a horizontal segment of that duration
returns to the start finish on distance 00

If a cyclist travels 1212 km home at 2424 km/h, the return takes 12/24=0.512/24=0.5 h, or 3030 minutes. Use that duration to place the final endpoint.

Do not join points before checking the event order and units. A visually plausible line is wrong if its endpoint time, distance or constant-rate gradient does not match the data.

Interpret gradients as speed and acceleration

A gradient is a rate of change: vertical change divided by horizontal change. Its meaning and units come from the graph axes.

Graph Gradient means Units example
distance–time speed km/h or m/s
speed–time acceleration m/s²
horizontal distance–time segment zero speed distance unit per time unit
horizontal speed–time segment zero acceleration speed unit per time unit

a=ΔvΔt=12−040−0=0.3 m/s2a=\frac{\Delta v}{\Delta t}=\frac{12-0}{40-0}=0.3\text{ m/s}^2

A negative speed–time gradient represents deceleration. When the question asks for the deceleration, report its positive magnitude unless a signed acceleration is requested.

For a curve, draw a tangent that touches at the required point and follows the local direction. Choose two well-separated points on the tangent—not necessarily on the curve—and calculate rise divided by run to estimate the instantaneous rate.

Do not use the height of a speed–time graph as acceleration: height is speed, while gradient is acceleration. Keep time and speed units consistent before dividing.

Find distance from the area under a speed–time graph

On a speed–time graph, area equals speed multiplied by time, so the area between the graph and the time axis is the distance travelled.

Linear section Area
rectangle base imesimes height
triangle frac12imesfrac12 imes base imesimes height
trapezium frac12imesfrac12 imes(sum of parallel sides)imesimes separation

Split the region at every change of gradient; label the time width and speed height of each rectangle, triangle or trapezium; convert units before multiplying; calculate each area; add all non-overlapping parts.

d=12(6+12)(30)+12(60)=990 md=\frac12(6+12)(30)+12(60)=990\text{ m}

Speed in m/s multiplied by time in seconds gives metres. Speed in km/h multiplied by minutes requires converting minutes to hours first.

After finding total distance, average speed is total distance/total time\text{total distance}/\text{total time}. It is not generally the arithmetic mean of the displayed speeds.

Cambridge limits these area calculations to linear graph sections. Avoid double-counting when subdividing, and do not use area under a distance–time graph as distance.