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Practice Problem in Physics for the JEE Main and Advanced
A stonc is dropped from a point $O$. There are points $P, Q, R$ along its paths II $O P, O Q, O R, \ldots$ are in
G.P., then prove the velocities at $P, Q, R, \ldots$ are in G.P.
Solution
Vel at $P-v_{1}-\sqrt{2 g h}, \quad$ Vel. at $Q-v_{2}-\sqrt{2 g h^{2}} ;$ Vel at $R-v_{3}-\sqrt{2 g h^{3}}$
since $O P, O Q, O R, \ldots \ldots \ldots . .$ can be taken as $h, h^{2}, h^{3}, \ldots \ldots \ldots \ldots .$
hence $v_{1}: v_{2}: v_{3}-v_{1}: v_{1} \sqrt{h}: v_{1} h-1: \sqrt{h}: h$
t.e., velocitics are also in G.P.