I am moving a tool over a varying distance. My motors can support constant acceleration and deceleration of $1 \space m/s^2$. I need to cover the distance as quickly as possible (accelerate for as long a time as the distance allows) and decelerate to $0 \space m/s$ at the end of the allowed distance.
Since the distance covered during deceleration varies depending on the velocity I accelerate up to; I'm looking for a good way to solve for both.
Thanks
If you want to travel between two points separated by a distance $d$, starting and ending at rest, and there is no limit on the maximum speed, then the shortest time is achieved by accelerating at maximum acceleration for the first half of the distance, and then decelerating for the second half.
The time taken to cover a distance $\frac d 2$ starting from rest at a constant acceleration $a$ is $t = \sqrt { \frac d a}$. So you accelerate for $\sqrt { \frac d a}$ seconds, reaching speed $\sqrt{ad}$, and then decelerate for $\sqrt { \frac d a}$ seconds. The total time taken is $2\sqrt { \frac d a}$ seconds.
