MathematicsExam code: 0580

Real Life Graphs

Coordinate Geometry and Graphs

PractiseFlashcards

Reading motion off the graph

  • A speed–time graph puts speed on the vertical axis and time on the horizontal one, so it looks similar to a travel graph but says something quite different
  • Check the vertical axis label first, because reading one as the other makes every answer wrong
  • Here the gradient is the acceleration, found as the change in speed divided by the time taken
  • The units are the speed units divided by the time units, so metres per second against seconds gives m/s²
  • A line sloping up is speeding up, and a line sloping down is slowing down, which is called deceleration
  • A horizontal line means constant speed, and a horizontal line at zero means the object is at rest
  • A straight sloping line means the acceleration is constant, while a curve means it is changing
A curve rising ever more steeply with tangents drawn at two points A and B, each with its own dashed right-angled triangle: the triangle at B is much steeper than the one at A, so the rate of change is greater there
Source: Rates of change of graphs by Save My Exams

Distance from the area

  • The distance travelled is the area between the graph and the time axis
  • On this specification the area you are asked for lies under straight sections of the graph only, so it splits into rectangles, triangles and trapeziums
  • Work out each piece separately and add them, or treat the whole shape as one trapezium where that fits
  • Check the units before writing the answer down, since a question may want the distance in kilometres when the axes are in metres and seconds

Worked example

Acceleration and distance from a speed–time graph

The graph shows the first 20 seconds of a car journey. Find the acceleration over the first 6 seconds, and the total distance travelled.

A speed-time graph with speed in metres per second up the side and time in seconds along the bottom: the line rises steadily from the origin to 9 m/s at 6 seconds, stays level at 9 m/s until 15 seconds, then falls steadily to zero at 20 seconds
Source: Speed-time graphs by Save My Exams

Solution:

  • Over the first 6 seconds the speed rises from 0 to 9 m/s
  • Acceleration = change in speed ÷ time = 9 ÷ 6 = 1.5 m/s²
  • For the distance, find the area between the line and the time axis, splitting it into three pieces
  • The first piece is a triangle: ½ × 6 × 9 = 27 m
  • The middle piece is a rectangle: (15 − 6) × 9 = 9 × 9 = 81 m
  • The last piece is a triangle: ½ × (20 − 15) × 9 = ½ × 5 × 9 = 22.5 m
  • Total distance = 27 + 81 + 22.5 = 130.5 m
  • The whole shape is a trapezium, so it can be done in one step as a check: ½ × (20 + 9) × 9 = 130.5 m

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