## Getting a Straight Line Graph

A straight line takes the form
$y=mx+c$
. You may think that if two quantities are not linearly related it is impossible to get a straight line graph between them. THIS IS NOT TRUE! It is often possible to transform variables and plot transformed variables to get a straight line graph.
Example: The thin lens equation is
$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$
.
We can rewrite this as
$\frac{1}{v}=- \frac{1}{u}+\frac{1}{f}$
. and plot
$\frac{1}{u}$
on the
$x$
axis against
$\frac{1}{v}$
on the
$y$
axis. The
$y$
intercept will be
$\frac{1}{f}$
and the gradient will be -1.
Example: The equation of radioactive decay is
$N=N_) \times e^{-\lambda t}$
. We can transform this by taking logs, obtaining
$ln(N)=-\lambda t + ln(N_0)$
. Now we can get a straight line by plotting
$ln(N)$
on the
$y$
axis against
$t$
on the
$x$
axis. The gradient will be
$- \lambda$
and the
$y$
intercept will be
$ln(N_0)$
.
Example: Boyles Law for an ideal gas is given by
$pV=CONSTANT$
where
$p, : V$
are the pressure and volume respectively of an ideal gas. We can write this equation as
$p=\frac{CONSTANT}{V}$
and plot
$p$
on the
$y$
axis against
$\frac{1}{V}$
on the
$x$
axis. The
$y$
intercept will be 0 (the line will pass through the origin) and the gradient will be
$CONSTANT$
.