Decelerating a Constantly Accelerating Ship

[zeframcochrane]

This is simply a thought experiment.

Say that I had a spacecraft capable of constantly accelerating or decelerating at a rate of 1g to 3g. I want to hit a target 20 light years away. How would I know where along the way to turn the ship around if I wanted to end at my destination decelerating at a rate of 3g. I have worked out that with a constant rate of acceleration (for example 1g) I would have to begin decelerating at 10 Lr. The catch here is that I want to change the rate of acceleration or deceleration gradually along the trip to eventually simulate 3g aboard the ship.

I've been building a dynamically modeled system to answer my question but there must be an easier way. I think I have to integrate my rate of acceleration over a set of points, but its been awhile since I took calculus. Any takers?

 

[PAllen]

The equations you need for any constant acceleration are given at:

http://www.desy.de/user/projects/Physics/Relativity/SR/rocket.html

For a case similar to yours where you assume constant 1g acceleration, followed by constant 1g deceleration, such that you end at rest 20 ly away in your starting frame, then you would reverse thrust after 3.023... years as measured on your rocket's clocks. Saying 10 light years as in your initial frame isn't useful because how would you identify this point in the rocket (unless there were a marker there)?

For varying acceleration, there is no avoiding calculus, and you may need numerical integration.