Equations of motion of a point sliding on a line of arbitrary shape

[PeroK]

Oh. True. Same mistake twice. So how could the conversion occur? Multiplying the result by the sinus and cosinus (y and x) of the angle formed by the tangent line of the shape to the horizontal?

A good exercise is to solve the motion on an inclined plane at angle $\theta$. The usual way by looking at tangential motion. Then, convert back to Cartesian coordinates.