Imagine racing your car against an F1 car. Your car has speed $100~km/h$ while the F1 car has speed $300~km/h$. The race director decides to show mercy and gives your car a head start. How long will it take for the F1 car to catch up to you?

Denote $t_0$ as the head start given to your car. Your car travels a distance $d$ as, \begin{equation} \label{eq:car} d = 100t, \end{equation} and the F1 car will travel that same distance as, \begin{equation} \label{eq:f1} d = 300(t - t_0). \end{equation}

Substituting \eqref{eq:car} into \eqref{eq:f1} and solving for $t$ gives \begin{aligned} 100t &= 300(t - t_0),
200t &= 300t_0,
\label{eq:t} t &= \frac{3}{2}t_0.
\end{aligned}

Now, let’s do a drag race! Let the drag strip be a $1~km$ long straight. What is the head start your car requires to tie or win the race?

Substituting \eqref{eq:t} into \eqref{eq:car}, we’ve \begin{equation} \label{eq:d} d = 150t_0. \end{equation} We plug in the numbers to get $t_0 = 1/150~h = 24~s$. Therefore, your car needs to have a head start of at least $24~s$ to tie or win.