A) The motion of the ball consists of two indipendent motions on the horizontal (x) and vertical (y) axis. The laws of motion in the two directions are:
![x(t)=v_0 \cos \alpha t](https://tex.z-dn.net/?f=x%28t%29%3Dv_0%20%5Ccos%20%5Calpha%20t)
![y(t)=h-v_0 \sin \alpha t - \frac{1}{2}gt^2](https://tex.z-dn.net/?f=y%28t%29%3Dh-v_0%20%5Csin%20%5Calpha%20t%20-%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2%20)
where
- the horizontal motion is a uniform motion, with constant speed
![v_0 \cos \alpha](https://tex.z-dn.net/?f=v_0%20%5Ccos%20%5Calpha)
, where
![v_0 = 8.00 m/s](https://tex.z-dn.net/?f=v_0%20%3D%208.00%20m%2Fs)
and
![\alpha=20.0^{\circ}](https://tex.z-dn.net/?f=%5Calpha%3D20.0%5E%7B%5Ccirc%7D)
- the vertical motion is an uniformly accelerated motion, with constant acceleration
![g=9.81 m/s^2](https://tex.z-dn.net/?f=g%3D9.81%20m%2Fs%5E2)
, initial position h (the height of the building) and initial vertical velocity
![v_0 \sin \alpha](https://tex.z-dn.net/?f=v_0%20%5Csin%20%5Calpha)
(with a negative sign, since it points downwards)
The ball strikes the ground after a time t=3.00 s, so we can find the distance covered horizontally by the ball by substituting t=3.00 s into the equation of x(t):
![x(3.00 s)=v_0 \cos \alpha t=(8 m/s)(\cos 20^{\circ})(3.0 s)=22.6 m](https://tex.z-dn.net/?f=x%283.00%20s%29%3Dv_0%20%5Ccos%20%5Calpha%20t%3D%288%20m%2Fs%29%28%5Ccos%2020%5E%7B%5Ccirc%7D%29%283.0%20s%29%3D22.6%20m)
b) To find the height from which the ball was thrown, h, we must substitute t=3.00 s into the equation of y(t), and requiring that y(3.00 s)=0 (in fact, after 3 seconds the ball reaches the ground, so its vertical position y(t) is zero). Therefore, we have:
![0=h-v_0 \sin \alpha t - \frac{1}{2}gt^2](https://tex.z-dn.net/?f=0%3Dh-v_0%20%5Csin%20%5Calpha%20t%20-%20%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2%20)
which becomes
![h=(8 m/s)(\sin 20^{\circ})(3.0 s)+ \frac{1}{2}(9.81 m/s^2)(3.0 s)^2=52.3 m](https://tex.z-dn.net/?f=h%3D%288%20m%2Fs%29%28%5Csin%2020%5E%7B%5Ccirc%7D%29%283.0%20s%29%2B%20%5Cfrac%7B1%7D%7B2%7D%289.81%20m%2Fs%5E2%29%283.0%20s%29%5E2%3D52.3%20m%20)
c) We want the ball to reach a point 10.0 meters below the level of launching, so we want to find the time t such that
![y(t)=h-10](https://tex.z-dn.net/?f=y%28t%29%3Dh-10)
If we substitute this into the equation of y(t), we have
![h-10 = h-v_0 \sin \alpha t- \frac{1}{2}gt^2](https://tex.z-dn.net/?f=h-10%20%3D%20h-v_0%20%5Csin%20%5Calpha%20t-%20%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2%20)
![\frac{1}{2}gt^2+v_0 \sin \alpha t -10 =0](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2%2Bv_0%20%5Csin%20%5Calpha%20t%20-10%20%3D0%20)
![4.9 t^2 +2.74 t-10 =0](https://tex.z-dn.net/?f=4.9%20t%5E2%20%2B2.74%20t-10%20%3D0)
whose solution is
![t=1.18 s](https://tex.z-dn.net/?f=t%3D1.18%20s)
(the other solution is negative, so it has no physical meaning). Therefore, the ball reaches a point 10 meters below the level of launching after 1.18 s.