Calculating Time to Reach Highest Point in Theoretical Mechanics Problem

[MerS]

Homework Statement

A small mass m is moving from the base-towards the top of a plane, leaned at an angle α to the horizontal base, with starting velocity Vo. There is friction with the plane and air friction force acting, and of course the gravitational force G. Find the equation for the mass' movement and the maximum height it will reach.

Relevant Equations

air friction force -> Fa=V*a (a is a constant)
plane friction force-> Fp=μG

A rough explanation of my attempts:
I set the coordinate axis in a way that x is the horizontal base and y is perpendicular. But I guess it is a 3d problem, so I struggle finding the right position of the coordinate system.
Then, my next step was applying 2nd NL for every axis. So far I have found the acceleration and the time the mass reaches the highest point, but I doubt its correctness and I am really getting nowhere.

 

[haruspex]

but I doubt its correctness

... and you'd like someone to tell you if it is. Can't quite see from here...

 

[phinds]

Draw a FBD and then we can discuss it meaningfully.

 

[MerS]

 

[haruspex]

View attachment 269506

At one place you have ##F_p=mg\sin(\alpha)##, at another ##F_p+mg\sin(\alpha)##.
You seem to have omitted drag.
You posted that you had found the time to reach the highest point. Where is the working for that?