A ball is released from a height h. If t₁ and t₂ be the time required to complete first half and second half of the distance respectively. Then, choose the correct relation between t₁ and t2 .A

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$t_{1} = (\sqrt{2}) t_{2}$

$t_{2} = (\sqrt{2} - 1) t_{1}$

$t_{1} = (\sqrt{2} - 1) t_{2}$

$t_{2} = (\sqrt{2} + 1) t_{1}$

answer is D.

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Detailed Solution

For first $\frac{h}{2}$
$\frac{h}{2} = \frac{1}{2} {gt}_{1}^{2}$
For total height h
$\begin{array}{l} h = \frac{1}{2} g {(t_{1} + t_{2})}^{2} \\ \frac{1}{\sqrt{2}} = \frac{t_{1}}{t_{1} + t_{2}} \\ 1 + \frac{t_{2}}{t_{1}} = \sqrt{2} \\ \frac{t_{1}}{t_{2}} = \frac{1}{\sqrt{2} - 1} \\ t_{2} = (\sqrt{2} - 1) t_{1} \end{array}$