# A stone is allowed to fall freely from rest. The ratio of the times taken to fall ….

Q: A stone is allowed to fall freely from rest. The ratio of the times taken to fall though the first metre and the second metre distance is
(a) √2 -1

(b) √2 + 1

(c) √2

(d) None of these

Ans: (b)

Sol: Using formula

$\displaystyle S = ut + \frac{1}{2}gt^2$

$\displaystyle 1 =\frac{1}{2}gt_1^2$

$\displaystyle 1+1 =\frac{1}{2}g(t_1+t_2)^2$

On dividing ,

$\displaystyle \frac{1}{2} = (\frac{t_1}{t_1+t_2})^2$

$\displaystyle \frac{1}{\sqrt 2} = \frac{t_1}{t_1+t_2}$

$\displaystyle \sqrt2 = \frac{t_1 + t_2}{t_1}$

$\displaystyle \sqrt2 = 1+\frac{t_2}{t_1}$

$\displaystyle \sqrt2 – 1 = \frac{t_2}{t_1}$

$\displaystyle \frac{t_1}{t_2} = \frac{1}{\sqrt2 – 1}$

$\displaystyle \frac{t_1}{t_2} = \frac{1}{\sqrt2 – 1} \times \frac{\sqrt2 +1}{\sqrt2 +1}$

$\displaystyle \frac{t_1}{t_2} = \sqrt2 + 1$