To solve the problem we can consider only the motion on the vertical direction (y-axis). Let's the take upward as positive direction, such that the acceleration of the motion (the gravitational acceleration) is negative: [tex]g=-9.81 m/s^2[/tex], because it is against the direction of the motion.
The initial speed of the ball on the y-axis is [tex]v_0 \sin \alpha[/tex], where [tex]v_0 = 8.8 m/s[/tex] and where [tex]\alpha [/tex] is the angle of the initial velocity v0 with respect to the horizontal, so it's the direction that we want to find.
The law of the velocity on the y-axis is:
[tex]v(t)=v_0 \sin \alpha + gt[/tex]
Where t is the time. At its highest point, the vertical velocity of the ball is zero, and this occurs when t=0.179 s: [tex]v(0.179 s)=0[/tex]. So we can write:
[tex]v(0.179 s)=v_0 \sin \alpha + gt=0[/tex]
And substituting the numbers we can find the value of the angle:
[tex]\sin \alpha = \frac{-gt}{v_0}= \frac{-(-9.81 m/s^2)(0.179 s)}{8.8 m/s}=0.20 [/tex]
[tex]\alpha = \arcsin (0.20)=11.5 ^{\circ}[/tex]
