# Straight line graphs

**D****rawing 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*, where

^{2}*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*for any case of constant acceleration from rest. Simple mathematics lead from the statement that Δv / Δt = acceleration, giving

^{2}*s = 1/2at*providing

^{2}*a*is constant. [ Δv = change of velocity, Δt = time taken.]

IF

*a*is constant, THEN

*s*=

*1/2at*because logic does that. So why might you plot the graph? To find out whether the trolley moved with constant acceleration.

^{2}