A ball is released from a height $h$. If $t_1$ and $t_2$ be the time required to complete first half and second half of the distance respectively. Then, choose the correct relation between $t_1$ and $t_2$.
Select the correct option:
A
$t_1=(\sqrt{2}) t_2$
B
$t_1=(\sqrt{2}-1) t_2$
C
$t_2=(\sqrt{2}+1) t_1$
D
$t_2=(\sqrt{2}-1) t_1$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
for first $\frac{\mathrm{h}}{2}$
$$
\frac{\mathrm{h}}{2}=\frac{1}{2} \mathrm{gt}_1^2
$$
For total height h
$$
\begin{aligned}
& h=\frac{1}{2} g\left(t_1+t_2\right)^2 \\
& \frac{1}{\sqrt{2}}=\frac{t_1}{t_1+t_2} \\
& 1+\frac{t_2}{t_1}=\sqrt{2} \\
& \frac{t_1}{t_2}=\frac{1}{\sqrt{2}-1} \\
& t_2=(\sqrt{2}-1) t_1
\end{aligned}
$$
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