I have built a Tchou-Tchou 'hyperloop' wagon that reaches 310 km/h on a 500 meter test rail. One of my investor customers wants to try the Tchou-Tchou, but I am a bit concerned about his safety. To impress him, the train is configured to reach 310 km/h with constant acceleration after 250 meters, then slow back down to 0 km/h with constant acceleration at 500 m.
Will he survive the best case scenario? And what is the best case scenario?
310 Km/h is 86 m/s. This means that on your 250 meters track (for acceleration), you'll have a mean speed of 43 m/s, meaning that you'll reach your 250m in 5.81 seconds. Now, 86 m/s reached in 5.81 s is 14.8 m/s², or about 1.5 g (same for deceleration). Maybe not really comfortable, especially for "regular" people not used to this kind of acceleration during transportation (except for rollercoasters), but undoubtely survivable.
1/2 a t^2 distance formula. 0.5 x 14.8 x 5.81^2 ~= 249.8 ~= 250.
– Kaz
Aug 05 '17 at 01:06
Tsts, it's easy peasy.
Colonel John Stapp made progressively harder and harder experiments with deceleration himself to find out what the human limits are (He advocated the safety belt, by the way).
Wikimedia, Public Domain
He stopped at December 10, 1954 with the rocket sled Sonic Wind from 1,017 km/h (632 mph) to zero in less than 1.4 seconds, experiencing a deceleration of nearly 46 g, meaning that the straps fixing him needed to hold the weight of an Indian elephant (Stapps weight was 77 kg (170 pounds), so the equivalent force was 3,5 t!).
For the more pragmatic people: The shorter the timeframe, the more g the human body can tolerate.
To expand on @a4android s Comment
For simpler numbers the following is calculated for a top speed of 100m/s
You can travel 500m in 10s with all of the following regimes:
constant acceleration of 20m/s2 for 5s then -20m/s for 5s. This solution has the lowest maximum acceleration/deceleration, but it has aprupt changes of the acceleration which are dangerous to the passenger.
Linearly increase the acceleration to 40m/s2 for 2.5s then linearly decrease to -40m/s2 for 5s then linearly increase to 0 for 2.5s. Here the acceleration is a continuous function without aprupt changes, but you need double the maximum acceleration. At around 4g this is still in the roller-coaster range.
The best is probably an intermediate solution, for example:
The correct formula to use here, given constant acceleration, is $v_f^2 - v_i^2 = 2ad$.
So $$a = \frac{v_f^2 - v_i^2}{2d}$$ with $$\begin{align} v_f &= 310\ \mathrm{km/h} = 86.11\ \mathrm{m/s} \\ v_i &= 0\ \mathrm{m/s} \\ d &= 250\ \mathrm{m} \end{align}$$ you get an acceleration of $14.83\ \mathrm{m/s^2}$ or about $1.50g$. This is well within human limits.
I have done some math...
The distance you travel while accelerating with constant acceleration is
$d= 1/2 a t^2$
while the velocity you reach in the same time is
$v = at$
since you state the distance and the velocity, we can solve it in acceleration and time.
$1/2 at^2 = 250$
$at = 86$
Which gives $a = 86^2/500 = 14.792 m/s^2$, almost exactly 1.5 g for a total of 12 seconds.
how do you get at=86?
— by using a fact "train is configured to reach 310 km/h". 310 km / h = 310'000 m / 3600 s → 86,1(1) m/s
– poige Aug 05 '17 at 08:39Top fuel dragsters currently accelerate from 0 to about 335 MPH (~540 KPH) in 1000 feet (~305 meters), taking about 3.5-4 seconds to do so (giving a little over 5 G's of acceleration). They then decelerate back to 0 in about another 5 seconds or so (around 3 G's of deceleration).
This is fairly impractical though. To do it, the cars use engines that produce around 10,000 horsepower. That puts enough wear and tear on the engine that it's standard practice to completely rebuild the engine every run.
Fighter jets can generate quite a bit more acceleration than that in a tight turn. Most have acceleration limiters, so they won't exceed about 8 G's, and will only maintain that for a very short time, then the jet will automatically "loosen" the turn to keep the pilot from passing out.
In this case, the acceleration as felt by the pilot is normally "downward"--i.e., pushing him/her down in the seat, rather that backward like the acceleration in a dragster. This has a significant effect--since it's pulling "downward", it's more difficult for the heart to pump blood to the brain. This leads to a "grey out" effect, where the brain (and eyes) are receiving little enough blood that vision becomes somewhat impaired.
Even achieving that takes fairly drastic measures--pilots wear "speed pants" to "squeeze" their legs, helping force blood upward instead of pooling in their legs. The "seat" in a modern fighter is also fairly reclined (e.g., around 30 degrees) to make it somewhat easier for the heart to pump blood to the pilot's head.
Getting to your actual question: these are probably close to the limit of what you can expect people to endure on a semi-regular basis. Accidents are often catastrophic, and even in the absence of catastrophic accidents the acceleration and deceleration take a substantial toll on drivers/pilots. A common injury among top fuel drivers is detached retinas. Don Garlits (top fuel driver, now retired) had surgery to fix a detached retina, and has admitted that it was fairly routine that the initial launch left him feeling "woozy" until he reached around the 300 foot mark.
So, getting to your specifications: accelerating at 1.5 G's should be no problem for any reasonably healthy adult. If you double that to 3 G's, there's still little likelihood of its being life threatening (especially given the relatively short track your postulating).
Tripling the acceleration to 4.5 G's gets you into the range where it's still entirely survivable, but you'd want to ensure the investor had a physical quite recently--it's getting to the point that you'd want to ensure that s/he was healthy enough rather than being able to take it for granted just because you didn't know of his/her being particularly unhealthy.
Top fuel dragsters can reach speeds of upwards of 400 km/h in under 3.2 seconds while traveling a distance of just 201 meters. They then decelerate quite rapidly using a combination of drag chutes and then wheel braking systems. The experience is no doubt extremely violent and uncomfortable, but drivers generally emerge from their cars unscathed.
This is a real world example using the fastest production automobile. It's not as fast as your requirements, but I think it gives some great information on the physics and power requirements to do what you want and the video link is entertaining.
The Bugatti Veyron has a top speed of 408.47 km/h (253.81 mph) and can go from 0 to 408 km/h to 0 in 90 seconds. Yes, it's going to require slightly more track than your Tchou-Tchou, but if you can do this in a production car, you really could do this in your hyperloop.
There are engineering challenges to consider.
You don't describe your top speed of your Tchou-Tchou. I assume it will do more than 100 mph. There are some parallels you should keep in mind to improve the story.
In the case of the Bugatti, it requires 250hp to achieve 100 mph. To achieve 253 mph, it needs another 750hp for a grand total of 1000hp. This means the engine powering the Tchou-Tchou needs to quadruple it's output just to make your challenge work.
Much of that is the resistance caused by the air in front of the car which creates friction and slows it down. You will have the same issues in a hyperloop tube because even if it's a vacuum and lower air pressure, it's going to be really hard to make complete vacuum. The biggest challenge with pushing what becomes in essence a ram down a tube is how to displace the air in front of the vehicle. In subways or rail tunnels, they build ventilation shafts to give the air somewhere to go besides forward. If you ride a subway regularly in an underground station, you already experience this to a degree when the air in the tube blows by you when a train is coming into the station.
Others covered stopping already, which is possible, tolerable, but maybe not so much fun.
Video of speed test
I really recommend the video. It's an engineer and television host explaining the engineering challenges of developing a car which is able to achieve significantly higher speeds than most vehicles.
Good luck.
