Three particles are projected vertically upward from a point on the surface of the earth with velocities $\sqrt{\left(\text{2gR/3}\right)},\sqrt{\left(\text{gR}\right)},\sqrt{\left(\text{4gR/3}\right)}$  respectively where R is the radius of the earth and g is the acceleration due to gravity on the surface of the earth. The maximum heights attained are respectively ${\text{h}}_{\text{1}}{\text{,h}}_{\text{2}}{\text{,h}}_{\text{3}}$

${\text{h}}_{\text{1}}=\frac{2\text{gR/3}}{2\text{g}-\frac{\text{2gR}}{\text{3R}}}$                ${\text{ h}}_2=\frac{\text{gR}}{2\text{g}-\frac{\text{gR}}{\text{R}}}$                                 ${\text{h}}_3=\frac{\frac{4\text{gR}}{3}}{2\text{g}-\frac{\text{4gR}}{\text{3R}}}$

$=\frac{2\text{gR/3}}{\text{4g}}$                                    $=\frac{\text{gR}}{\text{g}}$                                                     $=\frac{\text{4gR/3}}{\text{2g}}$

$=\frac{2\text{gR}}{12\text{g}}$                                  ${\text{h}}_2=\text{R}$                                                         $\text{=}\frac{\text{4gR}}{\text{6g}}$

${\text{h}}_{\text{1}}:{\text{h}}_3=\frac{2\text{gR}}{12\text{g}}:\frac{\text{4gR}}{\text{6g}}$

$=1:4$