How did we get that?

All truths are easy to understand once they are discovered; the point is to discover them.”
–Galileo Galilei

Do you remember the following equations of motion learnt in high school? These equations describe the motion of objects under constant acceleration:

$v = v_0 + at \qquad \qquad \qquad \boldsymbol{(1)}$
$x - x_0 = v_0t + \frac{1}{2} at^2 \qquad \;\boldsymbol{(2)}$
$v^2 = v_0^2 + 2as \qquad \qquad \;\;\;\boldsymbol{(3)}$

Take the second equation. Here v0 refers to the initial velocity. When we drop an object from some height above the ground, the initial velocity is zero. And (x – x0 ) refers to the distance travelled by the object, which is height, say ‘s’ . So, the equation can be written as:

$s = \frac{1}{2} at^2$

For a freely falling object the acceleration ‘a’ is ‘g’ ( the acceleration due to gravity). Replacing ‘a’ with ‘g’ and rearranging the factors in the above equation, we get this:

$t = \sqrt{\frac{2s}{g}}$