## 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 T^{2} 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 t^{2}, 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 t^{2}]

In fact we know that *s* is proportional to t^{2} for any case of constant acceleration from rest. Simple mathematics lead from the statement that Δv / Δt = acceleration, giving s = 1/2at^{2} providing *a* is constant. [Δv = change of velocity, Δt = time taken.]

IF *a* is constant, THEN *s* = 1/2at^{2} because logic does that. So why might you plot the graph? To find out whether the trolley moved with constant acceleration.