Got a difficult math calculus / differential equation problem for you to chew on.

Hypothetically, (Problem background) I’ve got a customer that wants line segments of glue applied to a surface with a constant width. When the glue machine dispenses, it dispenses at a constant rate (either on or off). The glue nozzle is on a gantry that can be commanded to move linearly from one coordinate to another. The gantry can be tuned to operate with S-Curve parameters: A maximum velocity (V), a maximum acceleration (A), and a maximum jerk (J - derivative of acceleration).

Question - what’s the distance D required for the gantry to travel to ramp up to its max velocity from stop, as a function of maximum jerk, maximum acceleration, and maximum velocity?

Hint: There are two closed form solutions, Under what conditions will the gantry reach maximum acceleration?