Straight line graphs
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
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.
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.