Question

A car, starting from rest, accelerate at the rate $f$ through a distance $s$, then continuous at constant speed for the some time $t$ and the decelerates at the rate $f/2$ to come to rest. If the total distance is $5s$, then prove that $s=\frac{1}{2}f{t}^2.$

Answer 1

$\begin{array}{c}s=0t+\frac{1}{2}f{t_1}^2\\ =\frac{1}{2}f{t_1}^2\to \left(1\right)\\ a=2s=v_{0}t\\ v_0=0+f{t}_1\\ 2s=f{t}_{1}t\\ t_1=\frac{2s}{ft}\\ s=\frac{1}{2}f{\left(\frac{2s}{ft}\right)}^2\\ s=\frac{2s^2}{f{t}^2}\\ s=\frac{1}{2}f{t}^2\end{array}$