1)A student at a window on the second floor of a high school building sees his math teacher coming along a walkway besides the building. He drops a water balloon from 18.0 m above the ground when it the teacher is 1.00 m from the point directly beneath the window. If the teacher is 170 cm tall and walks at a rate of 0.45 m/s, does the balloon hit his head? Does it him at all?

Question

thank you very much

The teacher's head is H = 18.0 - 1.7 = 16.3 m below the window from which the water balloon is dropped. The time required to fall that distance is
t = sqrt(2H/g) = 1.82 s.
The teacher moves horizontally
t*0.45 m/s = 0.82 m in that time. If he was 1.0 m behind the impact point when it was dropped, he will be 0.18 m behind the impact point when the water balloon reaches the height of his head.

That's pretty close. You need to consider the finite size of the water balloon and the person before deciding on an answer. The teacher will have moved even closer to the balloon before it hits the ground.

I would say the water balloon would barely miss his head but probably hit his forward-extended leg before it hits the ground.