From the top of a tower, a stone is thrown up and reaches the ground in time ${\text{t}}_{\text{1}}$ . A second stone is thrown down with the same speed and reaches the ground in time ${\text{t}}_{\text{2}}$ . A third stone is released from rest and reaches the ground in time ${\text{t}}_{\text{3}}$ then the relation between them is

h – Height of the tower

u – Velocity of projection of the stone

when stone is projected  vertically upwards. Taking vertically downwards motion of stone from the top of tower to the grand we have

$\text{u}=-\text{u,}\text{a}=\text{g,}\text{s}=\text{h}\text{and}\text{t}={\text{t}}_{\text{1}}$

$\text{S}=\text{ut}+\frac{1}{2}{\text{at}}^{\text{2}}$

$\text{h}=-{\text{ut}}_{\text{1}}+\frac{1}{2}\times \text{g}\times {\text{t}}_{\text{1}}^{\text{2}}\to \left(1\right)$

when stone is projected vertically down ward motion of stone from top of tower to ground we have,

$\text{u}=\text{u,}\text{a}=\text{g,}\text{s}=\text{h}\text{and}\text{t}={\text{t}}_{\text{2}}$

$\text{S}=\text{ut}+\frac{1}{2}{\text{at}}^{\text{2}}$

$\text{h}={\text{ut}}_{\text{2}}+\frac{1}{2}\times \text{g}\times {\text{t}}_{\text{2}}^{\text{2}}\to \left(2\right)$

when stone falls freely

$\text{u}=\text{0,}\text{a}=\text{g,}\text{s}=\text{h}$

$\text{h}=\text{0}+\frac{1}{2}{\text{gt}}_{\text{3}}^{\text{2}}\to \left(3\right)$

multiplying (1) by${\text{t}}_{\text{2}}$ and (2) by${\text{t}}_{\text{1}}$ and then adding we get

$\text{h}=-{\text{ut}}_{\text{1}}+\frac{1}{2}\times \text{g}\times {\text{t}}_{\text{1}}^{\text{2}}\times {\text{t}}_{\text{2}}$

${\text{t}}_{\text{2}}\text{h}=-{\text{ut}}_{\text{1}}{\text{t}}_{\text{2}}+\frac{1}{2}{\text{gt}}_{\text{1}}^{\text{2}}{\text{t}}_{\text{2}}$

$\underset{\_}{{\text{t}}_{\text{1}}\text{h}={\text{ut}}_{\text{1}}{\text{t}}_{\text{2}}+\frac{1}{2}{\text{gt}}_{\text{2}}^{\text{2}}{\text{t}}_{\text{1}}}$

$\left({\text{t}}_{\text{2}}+{\text{t}}_{\text{1}}\right)\text{h}=\frac{1}{2}{\text{gt}}_{\text{1}}{\text{t}}_{\text{2}}\left({\text{t}}_{\text{1}}+{\text{t}}_{\text{2}}\right)$

$\underset{\_}{\text{h}=\frac{1}{2}{\text{gt}}_{\text{1}}{\text{t}}_{\text{2}}\left({\text{t}}_{\text{1}}+{\text{t}}_{\text{2}}\right)}$

$\left({\text{t}}_{\text{1}}+{\text{t}}_{\text{2}}\right)$

$\text{h}=\frac{1}{2}{\text{gt}}_{\text{1}}{\text{t}}_{\text{2}}$

substitute ‘h’ value in equation (3) then

$\frac{1}{2}{\text{gt}}_{\text{1}}{\text{t}}_{\text{2}}=\frac{1}{2}{\text{gt}}_{\text{3}}^{\text{2}}$

 ${\text{t}}_{\text{1}}{\text{t}}_{\text{2}}={\text{t}}_{\text{3}}^{\text{2}}\Rightarrow {\text{t}}_{\text{3}}:\sqrt{{\text{t}}_{\text{1}}{\text{t}}_{\text{2}}}$