Introduction:
When a ball is dropped from a certain height, it falls due to the force of gravity. The time taken for the ball to fall depends on the height from which it is dropped. In this scenario, we have a ball dropped from 1m and 2m above a window, and we need to determine the time taken for the ball to cross the window in each case.

Analysis:
To analyze this scenario, we can consider the motion of the ball using the laws of motion and the concept of free fall. Let's break down the analysis into two parts:

Part 1: Ball dropped from 1m above the window
When the ball is dropped from a height of 1m above the window, it will undergo free fall. The key information we need is the time taken for the ball to reach the window.

Let's consider the equations of motion for an object in free fall:
1. The equation for displacement is given by: s = ut + (1/2)gt^2, where s is the displacement, u is the initial velocity, t is the time, and g is the acceleration due to gravity.
2. The equation for final velocity is given by: v = u + gt, where v is the final velocity.

In this case, the initial velocity (u) is 0 since the ball is dropped from rest. The displacement (s) is equal to the height of the window, which is 1m. The acceleration due to gravity (g) is approximately 9.8 m/s^2.

Using the first equation, we can rearrange it to solve for time (t):
1 = (1/2) * 9.8 * t^2
2 = 9.8 * t^2
t^2 = 2/9.8
t = sqrt(2/9.8)

Therefore, the time taken for the ball to cross the window when dropped from 1m above is T1 = sqrt(2/9.8) seconds.

Part 2: Ball dropped from 2m above the window
Similarly, when the ball is dropped from a height of 2m above the window, we can apply the same equations of motion.

Using the first equation:
2 = (1/2) * 9.8 * t^2
4 = 9.8 * t^2
t^2 = 4/9.8
t = sqrt(4/9.8)

Therefore, the time taken for the ball to cross the window when dropped from 2m above is T2 = sqrt(4/9.8) seconds.

Conclusion:
In conclusion, the time taken for a ball to cross a window depends on the height from which it is dropped. When dropped from 1m above the window, it takes T1 = sqrt(2/9.8) seconds, and when dropped from 2m above, it takes T2 = sqrt(4/9.8) seconds. This analysis is based on the laws of motion and the concept of free fall.