https://online-s.physics.uiuc.edu/cg.../13/fa04_1.gif

A projectile is shot out at 45o with respect to the horizontal with an initial velocity of vx = vy = vxy. When is the earliest time, t, that the velocity vector makes an angle of 30o with respect to the horizontal?
t = vxy / (g sin(30o))
t = vxy(1 - sin(30o)) / g
t = vxytan(30o) / g
t = vxy(1 - tan(30o)) / g
t = vxy / (g tan(30o))

I would do it as such:

When , then angle is given by



When the angle is 30, we get:



Hence, the ratio of the new v'y and vx is:



when the angle is 30 degrees.

Cross multiply, and since vx = vy;



When does v_y becomes v'_y?

Using v = u + at





EDIT: Mistake on the direction of g.

Unk: one minor correction: I believe that since the acceleration in the vertical direction is equal to -g, the last two lines should be:

Oh, duh, thanks, I missed that :o

I'll edit my post and correct it now.