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Real Life Graphs

Coordinate Geometry and Graphs

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 graph is made of straight sections, so the area 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
Exam tip

On a speed–time graph the gradient is the acceleration and the AREA is the distance. Reading the distance off the gradient, or the other way round, is the mistake this topic is built to catch, so check the vertical axis label before writing anything.

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