Vertical Fall?

Question

A steel ball is dropped from a building's roof and passes a window, taking 0.125 s to fall fromt he top to the bottom of the window, a distance of 1.20m. It then falls to a sidewalk and bounces back past the window, moving from the bottom to top in 0.125 s. Assume that the upward flight is an exact reverse of the fall. The time the ball spends below the bottom of the window is 2.00 s. How tall is the building?

Answer 1

The ball reaches the window with speed V and acceleration A (9.8m/s^2)

distance equation is
d = 1/2At^2 + Vt
(d-1/2At^2)/t = V
V = (1.20m - 1/2*9.8m/s^2*0.125s^2)/0.125s
V = 8.9875 m/s

To get to this speed the ball fell during time T such that
V = AT
T = V/A = 8.9875/9.8 = 0.9171s
during that time the ball fell from a height of
h = 1/2At^2 = 1/2*9.8*0.9171^2 = 4.121m

If the ball spends two seconds below the window, it fell for 1 second and rebounded for another 1s. In fact it reached the top of window with speed V and spent 1.125 second falling aftwerward so height below top of window is:

d = 1/2At^2 + Vt
d = 1/2*9.8*1.125^2 + 8.9875*1.125 = 16.3125m

So total height of building is height above top of window and height below top of window

h = 4.121m + 16.3125m = 20.43m

Answer 2

Let
t1 time to reach the top of the window
t2- time to reach the bottom of the window
t3 -to hit the ground
t4- to reach the bottom of the window

Since
S=0.5gt^2

S(window) = .5g[(t2)^2 - (t1)^2 )] or
S(window) = .5g[(t2) - (t1) ) (t2) + (t1)] = 1.2m
We know that
t2-t1= .125 sec (equation #1)
and now we have a second equation

(t2) + (t1)= 2 S(window)/ g [ (t2) - (t1) ]
(t2) + (t1)= 2 x 1.2 x 9.81/(.125)= 1.9572 sec (equation #2)

Solving equations for t1 an dt2 we have
t2=1.041sec and t1=0.9161 sec

Since we have a perfect elastic collision the motion from the ground mirrors the motion towards the ground so t3-t2=1 sec to prove we write
t3-t2=t4-t3 => t3=(t4+t2)/2 also
t3 - t2 + t4 - t3 = 2.0sec = t4-t2
t4 =2.0 + t2 = 2 + 0.9161= 2.92 sec

t3=(t4+t2)/2 = (2.92 +1.04)/2 = 1.98 sec

S= .5 g (t3)^2=19.2 m