A stone is dropped from the roof of a building. It is observed to...

Let us evaluate each statement by analyzing the motion of the stone.

For the motion across the window:
Displacement $s = 2.0\text{ m}$
Time $t = 0.2\text{ s}$
Acceleration $a = g = 10\text{ m s}^{-2}$

Using the second equation of motion to find the velocity at the top edge ($u$):
$s = ut + \dfrac{1}{2}at^2$
$2.0 = u(0.2) + \dfrac{1}{2}(10)(0.2)^2$
$2.0 = 0.2u + 5(0.04)$
$2.0 = 0.2u + 0.2$
$1.8 = 0.2u \Rightarrow u = 9\text{ m s}^{-1}$.
Thus, statement (I) is correct.

Velocity at the bottom edge of the window ($v$):
$v = u + at = 9 + 10(0.2) = 9 + 2 = 11\text{ m s}^{-1}$.
Thus, statement (II) is incorrect.

For the motion from the roof to the top edge of the window (initial velocity $= 0$):
Using $v^2 = u^2 + 2gh$, where $v = 9\text{ m s}^{-1}$ and $u = 0$:
$9^2 = 0 + 2(10)h$
$81 = 20h \Rightarrow h = 4.05\text{ m}$.
Thus, statement (III) is correct.

Time taken to reach the top edge from the roof:
Using $v = u + gt$
$9 = 0 + 10t \Rightarrow t = 0.9\text{ s}$.
Thus, statement (IV) is incorrect.

Exactly two statements, (I) and (III), are correct.

Answer: Two