Maximum Height

The maximum height, ymax, can be found from the equation:

vy 2 = voy2 + 2 ay (y - yo)

yo = 0, and, when the projectile is at the maximum height, vy = 0.

Solving the equation for ymax gives:

ymax = - voy2 /(2 ay)

Plugging in voy = vo sin(q) and ay = -g, gives:

ymax = vo2sin2(q) /(2 g)

where g = 9.8 m/s2

Note that the maximum height is determined solely by the initial velocity in the y direction and the acceleration due to gravity. It's not affected by what's happening in the x direction.