# How can I calculate a brachistochrone-like transfer burn with a changing acceleration?

I'm working on a sci-fi story with a few locations set across the solar system, and I want to make sure that it works as accurately as possible. In order to hop from planet to planet, the ships use energy beamed from a network of satellites across the system to continuously push them; cutting travel times down to only days or weeks per trip, removing the need for the ship to carry fuel mass, and generating artificial gravity on board in the process.

However, rather than just simply burning at a constant rate, I want the acceleration of the drives to rise and fall throughout the duration of the flight so passengers could adapt to the change in gravity when arriving at another planet. For example, a ship leaving Earth would start its transfer at 9.8 m/s of acceleration, but over the course of a few days the satellites would "throttle down" the force applied on the ship so that by the time it arrived at Mars, it would only accelerating at 3.7 m/s.

Also, somewhere in the middle of the flight there would have to be a point where the ship would have to stop its thrust, take 20 seconds to turn around, and continue thrusting in reverse in order to slow down. But even with this direction change, I want it to stop and resume it's change in acceleration like nothing happened (it would account for the gap in between, but it wouldn't need to alter the rate of change of acceleration afterwards).

Can someone please show me how I can calculate such a flightpath?

• 1. Are your readers really going to want to know the details of this calculation? If Apollo 13 had included all the details of how the flight plan was calculated, it would be far longer and far less popular a film. 2. If you don't know how to do this calculation, how many of your readers are going to quibble if you've gotten it wrong? – sammy gerbil Oct 23 '17 at 11:23
• @sammygerbil I've always been a fan of "hard sci-fi" novels, and although I'm not gonna show the calculations in the book, it would be nice to get the numbers right if someone was mentioning how long the trip would take, or when the ship would have to flip around. (also, trying to avoid spoilers as much as I can, there is a critical moment in the novel that involves a ship missing it's flip point.) – Mattias Oct 23 '17 at 14:49
• I think your requirements are not clear. I understand that you want to minimise the time for the journey, but you have not specified the maximum acceleration you want the passengers to experience, and also the maximum 'jerk' and 'jounce' - these are the rates at which acceleration is changing. ... The calculation is going to be very difficult. – sammy gerbil Oct 25 '17 at 11:42