\[y=mx+c\]
. 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 \[CONSTANT\]
.