A stone is thrown vertically upward with an initial velocity v₀. The distance travelled in time $\frac{1.5{\mathrm{v}}_0}{\mathrm{g}}$ is

Initial velocity = V₀

Time $=\frac{1.5{\mathrm{V}}_0}{\mathrm{g}}$

acceleration = -g

$\mathrm{S}=\mathrm{ut}-\frac{1}{2}{\mathrm{gt}}^2$

$\mathrm{S}={\mathrm{V}}_0\frac{1.5{\mathrm{V}}_0}{\mathrm{g}}-\frac{1}{2}\mathrm{g}{\left(\frac{1.5{\mathrm{V}}_0}{\mathrm{g}}\right)}^2$

$\begin{array}{l}\mathrm{S}=\frac{1.5}{\mathrm{g}}{\mathrm{V}}_0^2-\frac{2.25{\mathrm{V}}_0^2}{2\mathrm{g}}\\ =\frac{3{\mathrm{V}}_0^2\mathrm{g}-2.25{\mathrm{V}}_0^2\mathrm{g}}{2{\mathrm{g}}^2}\\ =\frac{0.75{\mathrm{V}}_0^2\mathrm{g}}{2{\mathrm{g}}^2}\\ =\frac{0.75{\mathrm{V}}_0^2}{2\mathrm{g}}\end{array}$