Calculating G-Force and Velocity

Simplifying a larger problem.

Building a sci-fi text game and I don't want the science to be too wrong. Imagine I have two points in space which are 111,125 kilometers apart and it takes 17 minutes and 41 seconds (1,061 seconds) to travel between them. Assuming constant acceleration and we flip halfway through burn, it seems to me that the g-force (relative to Earth's gravity) would be calculated as $$g_{earth} = \frac{meters /(\frac{seconds}{2})^2}{9.8}$$

With the above math, I have roughly 40gs (potentially killing the passengers), but I'm unsure if that is right and I don't know how to calculate max velocity in this scenario.

(Suggestions on best ways of surviving high-G burns also welcome).

• I suggest physics.stackexchange.com for most of this question. Commented Sep 14, 2016 at 10:00
• Mołot, thanks for the suggestion. I'll post it as soon as the "you can only post once every 40 minutes" ban is up :) Commented Sep 14, 2016 at 10:10
• By the way, your calculations seems right (I have a fever now so I may be mistaken, but they seem accurate, with Vmax over 200 km/s). So maybe just ask worldbuilding part, about mitigating the effects and making it survivable? Commented Sep 14, 2016 at 10:16
• Tip: graph it. See if the flightplan adds up in terms of velocity and position at each time, as a way to check the sanity of how you set up the problem. Commented Sep 14, 2016 at 11:10
– user
Commented Sep 14, 2016 at 18:19

As we on WB then let it be about more about practical aspects of that question, then just exercising in 5-grade math.

Pulling G: Human Responses to High and Low Gravity, https://books.google.ru/books?id=8l-kRUbT6_QC&pg=PA72 , page 72 and 73
$\tiny \text{as it states to be copyrighted material, I'm exercising my MIT copyrighted typing and story telling}$

Several experiments were conducted in Jhonsville, USA, by NACA (agency preceding NASA), in years 1950-1957 in studying effects of acceleration.

With body supporting couches, they achieved 20.7g peak for short time, 6 seconds, with 2 different testing human subjects.

They tried also water submerged support, project called Iron Maiden - 32g for 25 seconds(less than that, link below), reported mild sinus pain after.
Object name Gray, wished to test 40g acceleration in that setup, but 32g was maximum possible acceleration with that setup.

I assume, liquid-breathing was not used at that time, and as general idea seems pretty obvious and comes in mind to people who were not aware of such tests in the past (this, that, me included)

I was about to say that it states that main concern with lungs, they seems to be ok even without using liquid-breathing, but I'm a bit suspicious about that time 25 seconds of expression to that acceleration when they wished to go even for 40g, they could use saline solution.

This part of the story, as also a bit earlier experimentations, is described in NASA history page http://history.nasa.gov/SP-4201/ch2-4.htm which is more informative for us, cite:

In 1956, R. Flanagan Gray, a physician at the Johnsville laboratory, designed an aluminum centrifuge capsule that could be filled with water and was large enough to hold a man. After some initial troubles installing the contraption on the centrifuge and perfecting an emergency automatic flushing mechanism, the "Iron Maiden," as it was rather inaccurately nicknamed, went into use. In March 1958, Gray, immersed to his ribs in a bathtub-like device developed at the Mayo Clinic during the Second World War, had endured 16 g of headward (head to feet) acceleration. Then, the next year, Gray enclosed himself in the Iron Maiden and, positioned backward to the center of rotation and immersed in water above the top of his head, held his breath during the 25-second pattern to withstand a peak of 31 g transverse acceleration for five seconds. This performance with the water-filled aluminum capsule established a new record for tolerance of centrifuge g loads.

Interestingly enough are 2 moments here - direction of force (head to feet), and lung situation (held his breath), again do not states that saline solution was not used, although considering possible profile and time of exposure, goals of testing(reentry problem), initial partial submerging(immersed to his ribs) high likely it was not used.

Although even if it is not clear was the saline solution used or not, it is clear that liquid-breathing could be used to solve lung problem at high g accelerations. Concerns about more delicate human organ like brain are clearly out of scope that NASA history page, but they conducted shock tests with different subjects and for some reason is conclusion: judged that 83 g represented about the limit of human tolerance for deceleration.

Time which constant acceleration profile, with halfway acceleration and half way deceleration

$$S=\frac{at^2}{2} \Longrightarrow t=\sqrt{\frac{2S}{a}}, a=\frac{2S}{t^2}$$

in the case of equal parameters of acceleration and deceleration $S=\frac{1}{2}S^\prime$ where $S^\prime$ is 111'125 kilometers, and $t$ is half of time because of equal time of acceleration and deceleration process.

this way 40.25g is a correct answer.

As was mention above results shown by NACA this force(40g) is probably what human can tolerate even with relatively low tech solutions.

Time in conducted tests is considerably less than 1061 seconds, although liquid breathing could be one of the ways to solve the biggest oblivious problem with lung integrity. Probably few operations might be necessary for a longer time high g exposure, but it is not obvious if it is necessary for relatively short exposures like 20 min.

Another concern is cell organelles separation, but it starts from few hundreds of g for that time, not a concern here.

My conclusion would be, it is not only possible to withstand 40g, but relatively easy to achieve such resistance. My expectations are at least a few times higher in terms of the acceleration value, and way much higher in terms of time - with the advancing of technology in solving high g acceleration problem. Water/liquid is not the only way to solve the problem, there are other ways as reinforcing body part which is under stress, replacing water and liquid breathing with smart materials and probably other ways to achieve good results.

For humans I'm expecting:
100g expecting at least that value, will be very surprised, if not possible
not expecting 1000g, although will be nicely surprised, if it is possible
definitely not expecting 10000g with any matter based solution, will be very very surprised(and suspicious is that cheat or not).

• values, level of surprise is for actively functional organism, without cryogenic solutions and such, as they probably out of scope of OP's question - because of relatively short time (although it might be possible to freeze and de-freeze human in minutes or less(maybe even seconds), it is more the question of this procedure should do no harm and be compatible with live)
• For those who may not be aware, liquid breathing has not been demonstrated in people. Fair game for Sci-Fi, but as it's just mixed in with facts on this post I wanted to clarify that it's not currently demonstrable. +1 for all the research and math though. Commented Sep 16, 2016 at 16:48
• Also, there was a reason the experiments conducted were all for short periods of time beyond apparatus capability. Cushioning devices help for a short period of time, but eventually they will be fully compressed. Much over the time they have used those values are probably reached, at which point you might as well be stuck up against a steel wall. Commented Sep 16, 2016 at 16:49
• @GrinningX anatomical cushioning is used until nowadays in soyuz space capsule. With liquid breathing there are oblivious difficulties like circulation and damages to alveolus in lungs, but it is just one possible solution. And as example it is just 2in1 breathing and preventing something like pulmonary edema, but it do not have to be 2in1, it could be 2 separate solutions for oxygen exchange and fixing lungs. But thanks for bringing up those points. Commented Sep 17, 2016 at 23:28
• The Soyuz does not exceed ~4.5G... 1/10 or less of what you're talking about. I'm not sure, but I also doubt they experience even that force as long as you suggested a person could. At a point the maximum compression of the material will be reached, and then it will be like being held against steel. Commented Sep 21, 2016 at 13:53