The acceleration curve you describe would have infinite jerk at the start, stop and halfway points. A |jerk| minimizing curve for a fixed travel time would gra

The acceleration curve you describe would have infinite jerk at the start, stop and halfway points. A |jerk| minimizing curve for a fixed travel time would gradually (linearly) increase to maximum acceleration during the first 1/4 of the trip, gradually decrease back to no acceleration over the next 1/4, gradually increase to full deceleration over the next 1/4, and finally gradually decrease to no deceleration over the last 1/4.