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

v_y ² = v_oy² + 2 a_y (y - y_o)

y_o = 0, and, when the projectile is at the maximum height, v_y = 0.

Solving the equation for y_max gives:

y_max = - v_oy² /(2 a_y)

Plugging in v_oy = v_o sin(q) and a_y = -g, gives:

y_max = v_o²sin²(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.