The figure of the problem is missing: find in attachment.
(a) 1.64 s
The ball follows a projectile motion path. The horizontal displacement is given by
![x(t) = v_0 cos \theta t](https://tex.z-dn.net/?f=x%28t%29%20%3D%20v_0%20cos%20%5Ctheta%20t)
where
is the initial speed
t is the time
is the angle below the horizontal
We can rewrite this equation as
(1)
The vertical displacement instead is given by
(2)
where
is the acceleration of gravity
Substituting (1) into (2),
![y(t) = -x(t) tan \theta - \frac{1}{2}gt^2](https://tex.z-dn.net/?f=y%28t%29%20%3D%20-x%28t%29%20tan%20%5Ctheta%20-%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2)
We know that for t = time of flight, the horizontal displacement is
![x(t) =50.8 m](https://tex.z-dn.net/?f=x%28t%29%20%3D50.8%20m)
We also know that the vertical displacement is
![y(t) = -45 m](https://tex.z-dn.net/?f=y%28t%29%20%3D%20-45%20m)
Substituting everything into the equation, we can find the time of flight:
![\frac{1}{2}gt^2=-y -x tan \theta\\t=\sqrt{\frac{2(-y-xtan \theta)}{g}}=\sqrt{\frac{2(-(-45)-50.8 tan 32.0^{\circ})}{9.8}}=1.64 s](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dgt%5E2%3D-y%20-x%20tan%20%5Ctheta%5C%5Ct%3D%5Csqrt%7B%5Cfrac%7B2%28-y-xtan%20%5Ctheta%29%7D%7Bg%7D%7D%3D%5Csqrt%7B%5Cfrac%7B2%28-%28-45%29-50.8%20tan%2032.0%5E%7B%5Ccirc%7D%29%7D%7B9.8%7D%7D%3D1.64%20s)
(b) 36.5 m/s
We can now find the initial speed directly by using the equation for the horizontal displacement:
![x(t) = v_0 cos \theta t](https://tex.z-dn.net/?f=x%28t%29%20%3D%20v_0%20cos%20%5Ctheta%20t)
where we have
x = 50.8 m
![\theta=32.0^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%3D32.0%5E%7B%5Ccirc%7D)
Substituting the time of flight,
t = 1.64 s
We find:
![v_0 = \frac{x}{t cos \theta}=\frac{50.8}{(1.64)(cos 32.0^{\circ})}=36.5 m/s](https://tex.z-dn.net/?f=v_0%20%3D%20%5Cfrac%7Bx%7D%7Bt%20cos%20%5Ctheta%7D%3D%5Cfrac%7B50.8%7D%7B%281.64%29%28cos%2032.0%5E%7B%5Ccirc%7D%29%7D%3D36.5%20m%2Fs)
(c) 47.1 m/s at 48.8 degrees below the horizontal
As the ball follows a projectile motion, its horizontal velocity does not change, so its value remains equal to
![v_x = v_0 cos \theta = (36.5)(cos 32.0^{\circ})=31.0 m/s](https://tex.z-dn.net/?f=v_x%20%3D%20v_0%20cos%20%5Ctheta%20%3D%20%2836.5%29%28cos%2032.0%5E%7B%5Ccirc%7D%29%3D31.0%20m%2Fs)
The initial vertical velocity is instead
![u_y = -v_0 sin \theta = -(36.5)(sin 32.0^{\circ})=-19.3 m/s](https://tex.z-dn.net/?f=u_y%20%3D%20-v_0%20sin%20%5Ctheta%20%3D%20-%2836.5%29%28sin%2032.0%5E%7B%5Ccirc%7D%29%3D-19.3%20m%2Fs)
And it changes according to the equation
![v_y = u_y -gt](https://tex.z-dn.net/?f=v_y%20%3D%20u_y%20-gt)
So at t = 1.64 s (when the ball hits the ground),
![v_y = -19.3 - (9.8)(1.64)=-35.4 m/s](https://tex.z-dn.net/?f=v_y%20%3D%20-19.3%20-%20%289.8%29%281.64%29%3D-35.4%20m%2Fs)
So the impact speed is:
![v=\sqrt{v_x^2+v_y^2}=\sqrt{(31.0)^2+(-35.4)^2}=47.1 m/s](https://tex.z-dn.net/?f=v%3D%5Csqrt%7Bv_x%5E2%2Bv_y%5E2%7D%3D%5Csqrt%7B%2831.0%29%5E2%2B%28-35.4%29%5E2%7D%3D47.1%20m%2Fs)
While the direction is:
![\theta=tan^{-1}(\frac{v_y}{v_x})=tan^{-1}(\frac{-35.4}{31.0})=-48.8^{\circ}](https://tex.z-dn.net/?f=%5Ctheta%3Dtan%5E%7B-1%7D%28%5Cfrac%7Bv_y%7D%7Bv_x%7D%29%3Dtan%5E%7B-1%7D%28%5Cfrac%7B-35.4%7D%7B31.0%7D%29%3D-48.8%5E%7B%5Ccirc%7D)