How to I determine an equation for the maximum vertical position of an object?

The equation must be in terms of v_0 (initial velocity), theta (angle of launch), and g (gravity).





Thanks in advanceHow to I determine an equation for the maximum vertical position of an object?
Basically, you're asking for the derivation of the formula for maximum height of an object in two dimensional motion. Keep in mind that gravity acts as the y-component in motion in two dimensions. Instead of v_0, I'm going to use vi.





Start off with the following formulas:





vf = vi + at


0 = vi*sin(胃) - g*t


t = (vi*sin(胃)) / g





Now substitute these values into the formula for final position:





h = (vi*sin(胃)) * ((vi*sin(胃)) / g) - (g / 2) * ((vi*sin(胃)) / g)虏





h = (vi虏 * sin虏胃) / 2g





Now consider the horizontal (x) component of the object's motion:





R = (vi*cos(胃)) * 2t = (vi*cos(胃)) * ((2*vi*sin(胃)) / g)





R = (2*vi虏*cos(胃)*sin(胃)) / g





Using a trigonometric identity, the final answer is:





R = (vi虏 * sin(2胃)) / g