Some equations as first answers …
This is one of his recent interventions at the “International Astronautical Congress (IAC)” in Adelaide, Australia at the end of September where examples such as Paris NYC in 30 minutes or 12,000 km in 39 minutes or even Paris- Sydney in 50 minutes.
To do Paris-NYC in 30 minutes, we can always imagine the craziest solutions, the fact remains that Physics and Mathematics have held the course for several thousand years.
What distances, what speeds?
Physically, except earthquake, of a value that the Richter scale does not yet know, which would bring Paris and NYC closer to an order of magnitude at least, for the moment, the distance between the Apple and the City of Lights is of 5838 km. Five thousand eight hundred and thirty eight kilometers. It is not pifometric, it is mathematics that has taken over, with the formula of Haversine which allows you to calculate distances as the crow flies on a sphere.
To cover these 5,838 km in 30 minutes, you need an average speed of 11,576 km / h. And before obtaining this average speed, it is also necessary to reach a speed which is exactly the double if the acceleration is linear, that is to say the most flexible possible, that is to say that the path would consist of in a continuous acceleration of 15 minutes, then, for the other 15 minutes, the trip would be the equivalent deceleration.
It is therefore necessary to reach 23,352 km / h in 15 minutes.
What constraints?
This speed implies a (very) high atmosphere. Air resistance changes like the square of speed. At this speed, in a low atmosphere, between the friction forces, the heat they would release against the nose of the hull and above all, the energy that would have to be deployed to make any craft fly at this speed with such resistances , all this requires going to the upper atmosphere where particles are rare, probably around 1000 km altitude. This lengthens the journey by as much but above all, it further increases the speed: if the journey is now 7,500 km and no longer 5,838, it is no longer 23,352 km / h that must be reached, but 30,000! That’s 30% more. But always at the same time: no luck, the acceleration is proportional to the distance: the longer you have to travel for the same time in the end, the harder you need to accelerate.
To know the acceleration that allows you to do half of 5838 or 7500 in one acceleration a such that this distance is covered in 15 minutes, we can use the formula d = 1/2 at² and thus find that a = 2d / t² with aand m / s², d in meters and t in seconds.
So for 5838 km, we have a = [ 2 * (5 838 000)/2 ] / (900 * 900) = 7,2
That is to say that in the case “low to the ground”, each second, the speed increases by 7.2m / s compared to the speed of the previous second. 7.2 is not a trivial number: it is about 75% of the value of Earth’s gravity, a value known as 9.81 m / s / s. This means that during the 30 minutes that your trip lasts, your weight will increase by 75%: you will have 100% of your weight due to earth’s gravity and 75% more due to vertical acceleration.
– If you are 55 kg, you will have the equivalent of 41 kg more, and if you are 75 or 95 kg, you will have respectively 56 and 71 kg more!
For the most probable case where the journey is 7,500 km, the acceleration is 9.26 m / s / s, or about 95% of the earth’s gravity in addition.
So if you are 55, 75 or 95 kg, this time you will have to support 52, 71 or 90 kg more: roughly twice your own weight. And for 30 minutes, without any possible escape.
If we take the example of Paris-Sydney which is in fact worse in terms of distance / time ratio, the calculations give the following solutions:
For a Paris-Sydney in 50 minutes, i.e. 19,000 km (taking into account the high altitude), there, we have an acceleration of 11.7 m / s / s or 20% more than the earth’s gravitation (to always be taken into account in addition of course).
If you are 55, 75 or 95 kg, this time you have to support a overweight of 66, 90 or even 114 kg: you therefore weigh in total: 121 for 55, 165 for 75 and nearly 220 kg for the initial 95.
Imagine yourself, of good build, and spend 30 minutes sitting in a chair, with 65 kg on your knees and 50 kg on your shoulders.
Clearly, a Paris-Sydney in 50 minutes is simply impossible and to announce it so confidently is utterly ludicrous.
Other acceleration strategies?
So obviously, we can try to travel differently or take a break in the middle. But the price is immediately expensive: if we resume our Paris-NYC in 7,500 km, it is possible to accelerate a little shorter to enjoy part of the flight without acceleration.
If we take an acceleration of 12 minutes instead of 15 minutes, thus allowing a 6-minute break before deceleration, this time you take an additional 1g instead of 0.95g as initially calculated. The maximum speed is then 25,700 km / h instead of 30,000. On the first 3,750 kilometers, the acceleration this time concerns 2,550 km and the flight at constant speed is done over the following 2,400 km for 6 minutes. . Then, there are 12 minutes of deceleration at 1g (always with 1g of gravity in addition…): your weight doubles.
We can even try to think thatat the beginning, the acceleration will be weak because at constant thrust force the rocket is obviously very heavy of its fuel charge. And that on the second part of the acceleration, the thrust will be much stronger.
Let’s try to see what happens in our Paris-NYC where the second part of the acceleration is exactly twice the initial one (since at equivalent ejection speed, of the order of 4000m / s, there will be much less weight): initially, there will be an acceleration of 0.75 g for 7min30. This will give a speed of 3,300 m / s or 12,000 km / h. This will cover the first 750 kilometers. Clearly at this point it’s better. Except that: on the second part of the route (in time), 3000 km will still have to be covered by accelerating to 1.5g. The speed will increase to 9,950 m / s (36,000 km / h) and the kilometers before deceleration will be swallowed in 7min30 still, at the average speed of 24,000 km / h.
The fact remains that 1.5g of acceleration for 7min30 is 2.5 times your weight. It is therefore a completely mediocre result.
Note that even if the calculations are correct (often rounded for ease of reading) there are some approximations which do not fundamentally change the principle. For example, during the travel time in the upper atmosphere, the Earth’s gravity is less strong: at an altitude of 1000 km, it is not 9.81 but 7.33 m / s / s. The fact remains that whatever the travel position, either you will have a weight that will have taken 30 to 150% for 30 to 50 minutes minimum, or you will have to manage two accelerations along two axes, simultaneously. Which is not better.
It’s not just speeding up in life
Obviously, all of these calculations try to demonstrate the total discomfort of a Paris NYC in 30 minutes: in any case, you at least have your weight doubling.
But we could also talk about the purely economic aspect. From the ecological aspect but also from the purely local and occasional drawbacks: a rocket of the kind which would take several (tens of?) Passengers makes a noise on takeoff which is infernal even several kilometers away.
In NYC, putting a take-off barge requires it to be placed at least 50 kilometers from the shore: what about transport time? Boats in the Zone? Weather for the stability of the barge?
In Paris, as there is no sea, I would be personally curious to know where one can find a disc of 50 km of minimum diameter which is without resident, without classified fauna or flora etc. And even so, there would still be transit time to Paris. We would then have this magnificent paradox offered by Elon Musk: NYC → Paris in 30 minutes then aerocosmodrome → Paris center in 3 hours.
As long as it is not a day without a car!
The article originally appeared on Medium.
The contributor:
Vincent can be found on Facebook, Twitter and Medium.
–
