# How can you calculate a course based on constant acceleration burn and deceleration burn?

In the TV series The Expanse, a sci-fi space-engine called the "Epstein Fusion Drive" is a thruster capable of performing constant acceleration (up to and beyond "20g" (about $$200 m/s^{2}$$)) at a low fuel cost ("fusion particles"), in the scenes on-board in-flight ships, the characters seem to be in constant $$1g$$ (or similar), most likely from the constant thrust of their ships.

Given a distance of $$x$$ (in $$km$$), a mass of $$m$$ (in $$kg$$), and a max or "set" acceleration scalar of $$a$$ (in $$m/s^2$$, which is used during the acceleration and deceleration part of the flight, which for all intents and purposes of this question would be $$g = 9.80665$$), what would be a quick-n-dirty formula to calculate the time ($$t$$) taken from point a to b with a distance of $$x$$, ignoring all other possible forces and specificities, only this force that constantly accelerates, starting from a complete stop, and ending in a complete stop.

Additional notes: this question does not seem to answer this for me (and I don't think this question is a duplicate of it), as I cannot seem to make out this specific equation from its answer, my apologies if one of the answers has this exact answer, I'd probably then dont comprehend it good enough to understand it, and how it answers my question.

The equation you're looking for is:

$$T = 2 * \sqrt {D/A}$$

Where T is time in seconds, A is acceleration in m/s^2 (~9.81 for 1 g), D is distance in meters.

Note that this is dead-stop to dead-stop, whereas real interplanetary travel involves initial and final velocities which are usually very different from one another, and it also doesn't account for the motion of the destination body over the course of the flight, but it's okay for rough approximation purposes.

Project Rho is a good source for more information on more-or-less plausible science fiction engine technology.

• Thank you for your answer! This helped a lot with putting some of the time frames of the show in mind. ...putting in the distance from earth to jupiter (around 6 AU), assuming constant acceleration at 1g... wtf, thats just 7 days(!), no wonder the drive changed everything. Jan 5 at 20:19
• I learned how constant acceleration drives changed everything from the 1958 Heinlein novel Have Spacesuit, Will Travel which has an excellent explanation with examples that I could grasp as a kid. Too bad we don't know how to actually build them... Jan 5 at 20:59
• Basically you "just" need to build a fusion rocket that is scraping the upper theoretical limits of fusion power. I'd settle for a fusion rocket capable of 1G at all. Jan 6 at 10:23
• To clarify for "beginners," things like planets and Belter work-stations have large relative velocities, which is why, depending on the course taken, one might need lots more or lots less decel than accel. Jan 6 at 12:58
• This may be clear for people who have read the question, but caveat for anyone coming from google: the above formula is for straight-line acceleration. It treats acceleration from any source other than the engine (e.g. a body you're orbiting) as negligible. It will not apply to a constant low-thrust trajectory, like you might see from a present-day ion drive. Jan 9 at 0:29