TITLE: Fastest hyperloop-like vehicle which a human could endure
QUESTION [1 upvotes]: I'm currently writing on a science-fiction novel where a transportation system like the hyperloop has been built.
Now, I'm wondering how fast I can make this. Since the plot takes place on many different places around the earth it is crucial that the transportation is fast.
It should be possible to drive through Europe (Lissabon->Moscow, 4.500km) in about one hour.
My question is if there could be a Hyperloop that manages to do that (from a technical perspective) and would it be possible to accelerate that Hyperloop fast enough but such that a human would survive the drive?

REPLY [1 votes]: If the hyperloop track follows the curvature of the earth then the centripetal acceleration required to keep the passengers on the circular path is $a=v^2/r$, so $v=\sqrt{a_{max} R_{earth}}$. The Wikipedia page on g-force quotes the estimate that most people without training and special suits can handle a maximum of about $5$ g's of accelertation, or $5\times 9.8 \simeq 49 \text{ m/s}^2$. That gives a maximum speed of about $17 \text{ km/s}$, not including the forward acceleration required to get up to that speed.
If track is dug through the earth so that the track is straight the only concern is the linear acceleration required to speed up and slow down the vehicle. If we assume that the vehicle speeds up at a constant rate and then halfway through slows down at a constant rate the required acceleration is $a = 4d/t^2$. For your example this gives $a \simeq 1.39 \text{ m/s}^2$, or  $0.14 \text{ g}$, so covering 4,500 km in one hour is manageable.
If you combine the two effects for your example above you get $a_{max} = a_{linear} \times \sqrt{1+(d/R_{Earth})^2}\simeq 0.14 \text{ g}\times 1.22 = 0.17\text{ g}$. Still manageable.
If you put that all together and set $a = a_{max} \simeq 5 \text{ g}$ you can find the shortest feasible trip time for a given distance. For 4,500 km it is about 11 minutes.
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