A body is thrown vertically up with a velocity u, It passes three points A, B and C in its upwards journey with velocities $\frac{u}{2}$,$\frac{u}{3}$and $\frac{u}{4}$respectively. The ratio $\frac{AB}{BC}$ is:

$$\begin{gathered} -2\mathrm{g}\left(\mathrm{AB}\right)={\left(\frac{\mathrm{u}}{3}\right)}^2-{\left(\frac{\mathrm{u}}{2}\right)}^2 \\ =\frac{{\mathrm{u}}^2}{9}-\frac{{\mathrm{u}}^2}{4}=-\frac{5{\mathrm{u}}^2}{36} ...... \left(\mathrm{i}\right) \\ \mathrm{and}, -2\mathrm{g}\left(\mathrm{BC}\right)={\left(\frac{\mathrm{u}}{4}\right)}^2-{\left(\frac{\mathrm{u}}{3}\right)}^2 \\ =\frac{{\mathrm{u}}^2}{16}-\frac{{\mathrm{u}}^2}{9}=-\frac{7{\mathrm{u}}^2}{144} ........ \left(\mathrm{ii}\right) \\ \mathrm{Dividing} \left(\mathrm{i}\right) \mathrm{by} \left(\mathrm{ii}\right), \mathrm{we} \mathrm{get} \frac{\mathrm{AB}}{\mathrm{BC}}=\frac{20}{7} \end{gathered}$$