Related Rate of Change of an Angle

Hello

Out of pure maths interest, I was trying to solve the following problem:

We have: point O, anywhere on the plan; line AB, anywhere on the plan; point M, moving at speed described by func s(t) on AB, from A.

Find the rate of change of angle AOM (from time t), given that you can know everything about triangle ABO (lengths, angles...) but that this triangle is common (not iso, not rect, not equi....).

I have created a graph of this using geometric methods... with GeoGebra. It looks a bit like a Gauss curve.

When I now try to get the function for it, I tried several ways, and I couldn't. All I found was some strange graphs... (reversed U with vertical tangent on right side, graph in -1<y<1 -1<x<1; that was the latest... ?!)

I tried searching on internet but to no avail.

I know the basics of differentiation and am studying more.

If anyone has an idea...?

PS: I'll post a sketch when I've got time...