I haven't tried to work this out in detail, but I have a few concerns about what's been written.
1. The equation s = c tanψ doesn't seem to define a unique curve in terms of y vs. x. What are x and y in terms of ψ? I don't know if this matters.
2. Saying velocity is ds/dψ implies dψ/dt is constant. Is this the case?
3. You appear to be deriving the magnitude of the acceleration as the derivative of the magnitude of the velocity. But this is not valid. For example, for circular motion at constant speed, the magnitude of the velocity is unchanging, but the magnitude of the acceleration is non-zero.