A body travels a distance s in t seconds. It starts from rest and...

Portion OA, OB corresponds to motion with acceleration ' $\mathrm{f}$ ' and retardation ' $r$ ' respectively. Area of $\triangle O A B=S$ and $O B=t$. Let $O L=t_1$, $\mathrm{LB}=\mathrm{t}_2$ and $\mathrm{AL}=\mathrm{v}, \mathrm{S}=\frac{1}{2} \mathrm{OB} \cdot \mathrm{AL}=\frac{1}{2} \mathrm{t} \cdot \mathrm{v} ; \mathrm{v}=\frac{2 \mathrm{~S}}{\mathrm{t}}$ Also, $\mathrm{f}=\frac{\mathrm{v}}{\mathrm{t}_1}, \mathrm{t}_1=\frac{\mathrm{v}}{\mathrm{f}}=\frac{2 \mathrm{~s}}{\mathrm{tf}}$ and $\mathrm{r}=\frac{\mathrm{v}}{\mathrm{t}_2}, \mathrm{t}_2=\frac{\mathrm{v}}{\mathrm{r}}=\frac{2 \mathrm{~s}}{\mathrm{tr}} ;$\mathrm{t}=\mathrm{t}$_1+\mathrm{t}_2=\frac{2 \mathrm{~s}}{\mathrm{tf}}+\frac{2 \mathrm{~s}}{\mathrm{tr}}$ $t=$\left(\frac{1}{f}+\frac{1}{r}\right) \frac{2 s}{t} \Rightarrow t=\sqrt{2 s\left(\frac{1}{f}+\frac{1}{r}\right)}