A stone falls freely under gravity. It covers distances $${h_1},{h_2}$$  and $${h_3}$$ in the first 5 seconds, the next $$5$$ seconds and the next $$5$$ seconds respectively. The relation between $${h_1},{h_2}$$  and $${h_3}$$ is

Solution :
$$\eqalign{ & \because h = \frac{1}{2}g{t^2} \cr & \therefore {h_1} = \frac{1}{2}g{\left( 5 \right)^2} = 125 \cr & {h_1} + {h_2} = \frac{1}{2}g{\left( {10} \right)^2} = 500 \Rightarrow {h_2} = 375 \cr & \therefore {h_1} + {h_2} + {h_3} = \frac{1}{2}g{\left( {15} \right)^2} = 1125 \cr & \Rightarrow {h_3} = 625 \cr & {h_2} = 3{h_1},{h_3} = 5{h_1}\,\,{\text{or}}\,\,{h_1} = \frac{{{h_2}}}{3} = \frac{{{h_3}}}{5} \cr} $$