The maximum height, y_max, can be found from: v_y ² = v_y(0)² + 2 a_y (y - y(0)).

At maximum height, v_y = 0, while y(0) = 0. Solving for y_max gives:

ymax

=

- vy(0)2 2 ay

=

vo2sin2θ 2 g

Alternatively, use:  v_y(t) = v_y(0) - g t.    At the maximum:    t_max = v_y(0)/g

Substitute into y(t) = v_y(0) t - ½ g t² to give y_max = v_y(0)²/ 2g.

The maximum height is determined by: (i) the initial velocity in the y-direction, and (ii) the acceleration due to gravity. Motion along x is irrelevant!