They tell us that the cyclist starts braking in a constant manner \(\vec{x}_f\).

This means that she follows a constant acceleration motion, so her equation of motion is of the form

\begin{equation}
\vec{x}_f = \frac{1}{2} \vec{a} t^2 + \vec{v}_i t + \vec{x}_i,
\end{equation}

 

where  the final position is

 

\begin{equation}
\label{Cyclist_xfGeneral}
x_{fc} \, \hat{\textbf{i}} = – \frac{1}{2} a t^2 \, \hat{\textbf{i}} + v_{i} t \, \hat{\textbf{i}} + 0 \, \hat{\textbf{i}}.
\end{equation}

 

Notice that we know \(v_i\) (it is \(10\, \text{m/s}\)). But we do not know the acceleration or the time. So, in order to continue, we need more equations.

 

where is the final position

\begin{equation}
\vec{x}_f = \frac{1}{2} \vec{a} t^2 + \vec{v}_i t + \vec{x}_i,
\end{equation}