Straight line graphs
Teaching Guidance for 14-16
Drawing straight line graphs
Once you have plotted the points of a graph, checked for any anomalies and decided that the best fit will be a straight line:
- To select the best fit straight line, take a weighted average of your measurements giving less weight to points that seem out of line with the rest.
- Use a ruler to draw the line.
Interpreting straight line graphs
Proportionality: A straight line through the origin represents direct proportionality between the two variables plotted, y = mx. If the plotted points (expressing your experimental results) lie close to such a line, then they show the behaviour of your experiment is close to that proportionality.
Linear relationships: In many experiments the best straight line fails to go through the origin. In that case, there is a simple linear relationship, y = mx + c. Historically, one of the most far-reaching examples is the graph of pressure of gas in a flask (constant volume) against temperature. The intersect on the temperature axis gives an absolute zero of temperature, and an estimate of its value.
Identifying systematic errors: In some experiments, all measurements of one quantity are wrong by a constant amount. This is called a ‘systematic error’. (For example, in a pendulum investigation of T against l all the lengths may be too small because you forgot to add the radius of the bob. Plotting T2 against l will still give a straight line if every value of l is too short by the radius but the line does not pass through the origin.) In such cases, the intersect can give valuable information.
Checking for constancy: Consider the acceleration of a trolley. If you plot s against t2, where s is the distance and t is the total time of travel from rest, then you hope to get a straight line through the origin. [A straight line through the origin shows that s = constant t2]
In fact we know that s is proportional to t2 for any case of constant acceleration from rest. Simple mathematics lead from the statement that Δv / Δt = acceleration, giving s = 1/2at2 providing a is constant. [Δv = change of velocity, Δt = time taken.]
IF a is constant, THEN s = 1/2at2 because logic does that. So why might you plot the graph? To find out whether the trolley moved with constant acceleration.