## Getting a Straight Line Graph

\[y=mx+c\]

. 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 \[CONSTANT\]

.