I don't know if you can actually stitch the equations together though because they have different initial values, albeit in higher order derivatives than Newtonian mechanics cares about.

see https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... which was linked to from that new question:

> If we start at an arc length of 1/144 for example, it will run up the dome and arrive at the apex in 1 second. As we’ve seen, it has zero velocity and zero acceleration at this point, but moves off after anyway because it still has a positive value for snap.