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So I have decided that in my story on Mars, I'll have magnetic levitation trains. Now I'm calculating travel times and, considering passenger comfort, the question is:

How quickly could a passenger train accelerate while not causing significant discomfort? And how much time and distance would it take to achieve the top speed of 700 km/h?

The total travel distance is 1'000 km, in a straight line.

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    $\begingroup$ Comfort is kinda hard to quantify. Moscow metro trains manage ~1.3m/s<sup>2</sup> which is not much fun if you're standing up. This might be downright dangerous for someone who is used to Martian gravity which is a third of Earth's. Given the distance though, there's no need for excessive acceleration if everyone can be just a little bit patient. $\endgroup$ May 5, 2023 at 16:29
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    $\begingroup$ It also depends on the amenities. If everyone's in comfortable, well-padded forward-facing seats, that's a different value from people having to hold on to handrails while carrying a briefcase in their other hand. $\endgroup$
    – jdunlop
    May 5, 2023 at 17:23
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    $\begingroup$ @jdunlop Problem is that the train has to decelerate at the other end. At which point forward-facing seats are now the wrong direction for comfort. So unless your vehicle turns around in the middle (good for spacecraft, not for trains) best to keep the acceleration to a level that is OK forward or backward. $\endgroup$ May 5, 2023 at 20:41
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    $\begingroup$ Your question is really, "What is a tolerable acceleration for a passenger train?" After that, 700km/h is arbitrary and solved with simple math. $\endgroup$
    – user458
    May 6, 2023 at 0:02
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    $\begingroup$ @manassehkatz-Moving2Codidact - not at all; just have the seats rotate. $\endgroup$
    – jdunlop
    May 6, 2023 at 3:14

6 Answers 6

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About 27 kilometers over 4 minutes 45 seconds

Let's first answer the passenger comfort question.

Humans generally feel acceleration, not necessarily velocity. Wikipedia provides that the Shanghai Transrapid accelerates at 300 km/h (186 mph) in 2 minutes and 15 seconds, or approximately 0.617 m / s2. Incidentally, the Shinkansen N700 Series, a high-speed conventional train, accelerates at 0.720 m/s2. Between the two, we can presume the 0.70-0.75 m/s2 range as reasonable for human passengers.

The Newtonian equations of motion offer much in terms of helping you calculate the time and distance of a moving object. They are:

\begin{aligned}v&=at+v_{0}&[1]\\r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}&[2]\\r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t&[3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)&[4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}&[5]\\\end{aligned}

  • v is final velocity
  • v0 is initial velocity
  • a is acceleration
  • t is time
  • r is final distance
  • r0 is initial distance

For the purposes of this response, I will use this calculator. Using the numbers provided:

  • 0.6 m/s/s provides a displacement of approximately 31,500 meters and 324 seconds
  • 0.7 m/s/s provides a displacement of approximately 27,000 meters and 278 seconds
  • 0.8 m/s/s provides a displacement of approximately 23,600 meters and 243 seconds

By this calculation, about 27 kilometers over circa 4 minutes 45 seconds will be sufficient to comfortably accelerate to full speed.

EDIT: as some have mentioned, the Hong Kong MTR Rotem EMU claims an acceleration of 1 m/s/s. This calculation would yield 19 km over 194 secs (3 min 14 seconds). Ultimately given the nature of this question it is up to you which estimate to use specifically.

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    $\begingroup$ These acceleration values for the Shanghai Transrapid and the Shinkansen just tell me that the companies didn't bother with higher accelerations, not necessarily because it would be unbearable for human passengers, but maybe just because there was not much benefit to have a higher acceleration, and presumably a higher acceleration would be more expensive in terms of fuel, and in terms of engineering appropriate safety measures, and in terms of having comfortable seats and securely-stored luggage. $\endgroup$
    – Stef
    May 6, 2023 at 8:42
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    $\begingroup$ The transrapid and shinkansen appear to both have a top speed of 300km/h. Knowing that, there is probably not much incentive for the company to have an extremely high acceleration. Suppose you can come up with a mechanism so that your train accelerates to 300km/h in 1minute instead of 2minutes15. How much time and money would that save for the company? Probably not much, especially if this mechanism is itself very expensive. But the OP's Mars train has a top speed of 1000km/h. So there is incentive to have a higher acceleration. $\endgroup$
    – Stef
    May 6, 2023 at 8:44
  • $\begingroup$ And the gravity of Mars is about 38% the gravity of Earth, which might make it easier or harder to have a higher acceleration. $\endgroup$
    – Stef
    May 6, 2023 at 8:46
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    $\begingroup$ @Stef It depends on the mechanism on which the train operates. Since we are talking about a maglev, though, higher gravity will only result in higher energy requirements to keep the train levitating. It would not have an effect on how much energy is required to accelerate since that is only a function of the train's mass. $\endgroup$
    – Abion47
    May 7, 2023 at 3:29
  • $\begingroup$ @Stef I'm using the Transrapid and N700 values from (mostly) personal experience - the Transrapid has very little incentive for higher acceleration, but the N700 would benefit in some ways due to station proximity, etc. Qualitatively, taking the N700 for me seemed to be the upper bound of "comfort", personally. That said, it hearkens back to the somewhat subjective way the question was framed. $\endgroup$
    – dreamforge
    May 8, 2023 at 1:42
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Maximum comfortable train acceleration doesn't depend on the top speed. So, just look at the data of existing trains for a reference.

For example the Shinkansen

Another feature of the N700 is that it accelerates more quickly than the older 700 series Shinkansen trains, with a maximum acceleration rate of 2.6 km/h/s (0.72 $m/s^2$).

With that in mind, calculating how long does it take to reach a certain speed and to complete a given distance is trivial high school math on the known formulas

$v=a\cdot t$

$s=1/2 a \cdot t^2$

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Worldbuilding Meta, or in Worldbuilding Chat. Comments continuing discussion may be removed. $\endgroup$
    – L.Dutch
    May 8, 2023 at 9:43
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All the other answers use totally valid math (I guess).

But the passengers in your story are martians. They are used to a 3.72076 m/s2 gravity.

How this affect their tolerance to acceleration is ... whatever suits you until proven wrong.

Also earth newcommers may enjoy an increased gravity

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    $\begingroup$ Upvoted! If the inhabitants of Mars are human colonists from Earth, it would make sense that the low gravity of Mars is actually a problem for them, so they need to spend time in high acceleration environment, regularly, in order to maintain a healthy human body. This could actually be a very good incentive to build high-acceleration trains. $\endgroup$
    – Stef
    May 6, 2023 at 8:47
  • $\begingroup$ @Stef Good point. I was focusing on native martians but I udpate my answer $\endgroup$
    – Madlozoz
    May 6, 2023 at 10:03
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    $\begingroup$ @Stef Mandated weekly around-the-globe traintravel, I already love it! $\endgroup$
    – Hobbamok
    May 8, 2023 at 9:25
  • $\begingroup$ Using acceleration to produce artificial gravity, while good physics, here it's in the wrong direction. Unless they're in a pivoting cabin, so their feet are always pointed away from the acceleration vector (so it would reverse when switching from accelerating to decelerating), this wouldn't help the occupants physiologically. In theory, you could do it by having each row of seats pivot around an axis, with more mass on the foot end than the head end. Bear in mind that no matter how you did it, it would be impossible to traverse the length of the train (e.g., for the bathroom). $\endgroup$
    – Jeffiekins
    May 8, 2023 at 18:05
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Comfortable acceleration for humans on Mars

Gravity on Mars is about 38% of gravity on Earth.

If the inhabitants of Mars are human colonists, they probably want to be subjected regularly to an acceleration closer to that of Earth, in order to maintain a healthy human body and avoid health issues such as loss of bone; atrophy of antigravity muscles; decreased blood volume; loss of blood circulation, leading to intolerance to standing up. Reference: long-term effect of low gravity on the human body.

  • Gravity on Earth has magnitude: $g_E = 9.81 \,m/s^2$.
  • Gravity on Mars has magnitude: $g_M = 3.72 \,m/s^2$.
  • If your train accelerates with horizontal acceleration $a$, then the total force felt by the passenger will have magnitude: $$\sqrt{{g_M}^2 + a^2}$$

So now we can solve the equation $\sqrt{{g_M}^2 + a^2} = g_E$ in order to determine a very comfortable $a$.

This equation corresponds to the following diagram, in which the passenger in the train on mars feels a total acceleration with same magnitude as the acceleration of Earth gravity, although the acceleration will be oblique and closer to horizontal than vertical: Total acceleration: mars plus train equal earth

Solve for $a$: \begin{align*} \sqrt{{g_M}^2 + a^2} & = g_E \\ a & = \sqrt{{g_E}^2 - {g_M}^2} \\ a & = \sqrt{{9.81}^2 - {3.72}^2} & m/s^2 \\ a & = 9.08 & m/s^2 \\ \end{align*}

Time and distance to reach top speed of 700 km/h

From this comfortable value of $a$, you can deduce the time $t_0$ and distance $x(t_0)$ necessary to go from speed $v(0) = 0$ to speed $v(t_0) = 700\,km/h = 194.4 \,m/s$.

This is as simple as : \begin{align*} v(t) & = a t \\ x(t) & = \frac 1 2 a t^2 \\ \end{align*} Thus the time is: \begin{align*} t_0 & = \frac{v(t_0)}{a} \\ t_0 & = \frac{194.4}{9.08} & s \\ t_0 & = 21.4 & s \\ \end{align*} And the distance is: \begin{align*} x(t_0) & = \frac{1}{2} \, a \, {t_0}^2 \\ x(t_0) & = \frac{1}{2} \times 9.08 \times {21.4}^2 & m \\ x(t_0) & = 2083 & m \\ \end{align*}

The train will accelerate for about 21s, over about 2.1km.

Jerk

In practice you probably want the acceleration to change smoothly from 0 to max acceleration down to 0 again. A brutal change in acceleration is called a jerk and as the name suggests, it is quite unpleasant. A jerk of up to 1g/s (9.8 m/s³) shouldn't be harmful, but unpleasant. If this train is part of a military operation, or if the passengers are well-seated and the luggage is well-secured, you could theoretically reach top acceleration in one second. But if this is a passenger train where passengers expect a modicum of comfort, you should probably spread the jerk over a longer period. I would probably assume 30 seconds and 3 km instead of 21 seconds and 2.1 km.

Voluntary acceleration

I started my answer by stating that humans on Mars probably wanted to be subjected regularly to high acceleration. Some humans are probably willing to pay for that, even when they don't have to travel. However, with the acceleration of 9 m/s mentioned above, you only get 21s of acceleration before you reach the maximum speed of 700km/s.

To subject your humans to high acceleration for longer periods of time, I can think of two solutions:

  • a circular track, in which case you can have a constant acceleration, oriented in the radius of the circle;
  • alternate between high acceleration and high deceleration, in which case you need to rotate the seats of the passengers inside the vehicle, so they are always facing in a comfortable direction.
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  • $\begingroup$ The top speed is 700 km/h. You probably mistook the distance of 1000 km as top speed. $\endgroup$ May 6, 2023 at 11:57
  • $\begingroup$ @KrišjānisLiepiņš Indeed I did! I have edited my answer for a top speed of 700km/h. $\endgroup$
    – Stef
    May 6, 2023 at 12:06
  • $\begingroup$ Jerk isn't a brutal change of acceleration, it's a change of acceleration. $\endgroup$ May 11, 2023 at 8:43
  • $\begingroup$ @AmiralPatate Are you trying to argue that a brutal change of acceleration is not a change of acceleration? $\endgroup$
    – Stef
    May 11, 2023 at 18:12
  • $\begingroup$ No, I'm saying jerk is the time-derivate of acceleration, i.e. any change of acceleration. There's no notion of brutality to it. $\endgroup$ May 12, 2023 at 5:57
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There's a Tom Scott video https://youtu.be/4ZX9T0kWb4Y about a maglev train in japan.

He doesn't give the acceleration, but the train accelerates from 195km/h at 39.47km from the destination to 503km/h at 31.4km. Thats a speed difference of 308km/h in 8.07km.

If i calculated this correctly that's an acceleration of about 1.03m/s^2.

Tom Scott describes this as significant, but not unpleasant, so that seems a good guideline.

Of course, with some disregard for passenger comfort, or appropriate acceleration couches, you could certainly up this significantly, keeping in mind that martians will be accustomed to only a third of our gravity.

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    $\begingroup$ Acceleration is not measured in metres per second. Also the question doesn't just ask for the acceleration, but also the time and distance in order to achieve the given speed. $\endgroup$
    – kaya3
    May 6, 2023 at 4:31
  • $\begingroup$ yeah, sorry i missed a square. time and distance are obviously a function of the acceleration the OP is willing to use. and then they're a simple calculation away. $\endgroup$
    – ths
    May 6, 2023 at 11:20
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Affect of reduced Martian gravity

Mars has a surface acceleration of about 3.7 m/s/s - or about 38% of that on earth.
The acceleration figures used below and in most or all other answers are based on terran experience. The tolerable figures in Mars much lower gravity may perhaps be lower or even much lower.
Native born martians would never experience 1g in "walking about". If musculature and bone development were adapted to their local environment I'd expect martians to be tall, of thinner bone structure, and perhaps weaker due to typical object "weighing" far less.

Earthies staying for any period, or permanently, might be expected to also be less tolerant of higher g forces. For a good answer on this you need input from biological experts. Scaling figures given here and in other answers down in g by about 2.5 and up in time by 2.5 may be in order.


Time to speed can be determined form the formula

  • T = V / A
    (similarly) V = A x T

km/h = m/s x 3.6
So

  • Seconds = km/h /(m/s x 3.6)

So 700 km/h at 1 m/s/s = 700/(1 x 3.6)= 200 seconds.

Distance = 0.5 x acceleration x time_squared.
Here metres = 0.5 x 1 m/s/s x (200 seconds squared)
Distance to 700 km/h = 20 kilometres.

T is time in seconds
V is velocity in m/s
A is acceleration in m/s/s (metre per second per second)

km/h = m/s x 3.6
Terran acceleration due to gravity is about 10 m/s/s (closer to 9.8, but 10 is a useful figure)

If the above acceleration is scaled down to 0.4 m/s/s then time to speed is about 500 seconds, and distance to speed is about 125 kilometres ! Much further due to distance rising with time-squared.


This page from the "Tramway & Urban transport site says:

Modern trams have rapid acceleration – 1.3 m/s2 is often used – so to reach 50km/h on a level surface will take 11 seconds. To calculate the tractive effort required to attain this, we multiply the acceleration rate by the vehicle’s mass in tonnes. In our example, this gives 78kN, to which we add the rolling resistance to give 81kN – far greater than the 3kN required at a constant speed.

The Hong Kong MTR trains are the fastest accelerating ones I've met at a claimed 1 m/s/s maximum in normal use. Singapore are similar. I'd say from experience that the deceleration is higher - standing passengers need to brace against deceleration and maybe each other. This is well accepted as "part of the ride" and not seen as at all remarkable or uncomfortable.

Image form here Wikipedia - Mass Rapid Transit (Singapore)

enter image description here


I have travelled in the Shanghai Maglev / Transrapid on 'quite a few' occasions. It has a posted max speed of 430 km/h - well above the 300 km/h suggested elsewhere here.

Acceleration is "pleasant and noticeable".
Higher acceleration would not be unpleasant and would add to the experience, but would make walking in the aisles, which you can do at any time without restriction, more hazardous. Given the figures suggested by others for actual accelerations I feel that 1 m/s/s would be very comfortable.


In a system with short distances as well as longer ones, increased interstation acceleration would make a significant difference to trip times.
If passengers are seated accelerations well above 1 m/s/s would not be uncomfortable. Braking while facing in the forward direction would be more noticeable than acceleration which presses you into your seat.

Motor vehicle accelerations of several 10ths of a g are very much "part of the ride". If harnessed a 0.5g forward facing deceleration would be significant but bearable.

My late lamented MR2 sportscar long ago exceeded these levels in 2 and sometimes 3 dimensions. These levels may have been above "acceptable comfort levels" but for a belted in occupant were never painful.


Desired speeds and resultant power:

Circumference of Terra and Mars are close to 40,000 and 20,000 km respectively. Once there are enough population centres (say Kim Stanley Robinson Mars trilogy scenarios) then settlements will still be very widespread.
Distances of 1000 km would be liable to be common between clusters of population and the ability to support end to end trips of 5000-10,000 km highly desirable - if not always practical.
It's likely that IF rail could be fast and energy economic that rail could be the predominant transport mode. "Air"craft may be viable with eg thermal transfer engines but lack of atmospheric oxidant will make that hard. SO an eg 1000 km/h train could be immensely attractive.

The greatly less dense Martian atmosphere helps decrease power requirements due to atmospheric drag - the predominant energy sink at high speed.
Martian atmospheric density is about 0.02 kg/m^3 (as opposed to 1.2 at earth sea level so power at 1000 km/h (280 m/s) will be about
Power = Cd x 280^3 x 0.5 = 220 kW per square metre of frontal area at Cd of 1.
Assume Say Cd=0.3 and A of 10 m^2 giving power of 660 kW - say 1000 kW actual. This is for a "not too long " train - friction and body drag add with length. See

AERODYNAMICS OF HIGH-SPEED TRAINS - their use of Cd is unfamiliar to me. I'll try to resolve. My figure is useful as used.

How a 1000 km/h train would be propelled is very much "a problem for the student". The low density atmosphere would make aerodynamic propulsion difficult. Wheeled propulsion at 1000 km/h would be 'interesting'. MAGLEV could be attractive but the energy and rail construction issues are also challenging. (10,000 km of MAGLEV track is very far from trivial).


ENERGY SOURCES:

Kim Stanley Robinson's Mars trilogy Red / Green / Blue Mars provides some good ideas re Mars development.
This story's scenarios will differ, but things can be learned from KSR's tales. He had massive AI controlled machinery available. He operated near the edge of it can be done it will be done. They made synthetic diamond to build space elevators when needed. They ... !!!

Time period for this story is unknown.

Mars has substantial Uranium and Thorium in some areas.
Mars had/(has?) natural reactors that bred U233 and went critical and ...!!!
Importation of U/Th is feasible.
Construction of reactors on planet is highly feasible longer term.
KSR's people used Rickover reactors - a hat tip to US Navy Admiral Rickover. – See ref at end to "Rickover reactor" for lots of interesting real world stuff.

Mars has massive volcanic activity (but no tectonic plates). Potential energy is large - but not much good for transport.

Solar insolation is far lower than earth's - about 100-200 Wh/m^2 peak compared to Terra's 1 kW/m^2 typical. Daily kWh/m^2 is typically 1.5-2.5 which is better than I'd have expected.

An electric train is OK over 100s of km. Over 1000's of km with little parallel infrastructure is harder. May still be viable. Very high voltage allows long distance feeds at low loss.

Long term, nuclear seems attractive - with maybe thermal propulsion despite the thin atmosphere.

Elon and others propose Methane synthesis for rocket fuel.

KSR's people drill very very very deep "Moholes" to tap thermal energy.

Martian wind is high velocity but low energy, mostly.

KSR has vast water reserves in aquifers just waiting to green the planet. One can hope :-). Copious water does seem to exist. But not that much. Probably. –

_______________________________________________

EVIDENCE FOR A LARGE, NATURAL, PALEO-NUCLEAR REACTOR ON MARS.

Wikipedia - Ore resources on Mars

Solar Radiation on Mars NASA 1989

Hyman Rickover his reactors may yet be seen on Mars :-)

Rickover Reactor - Google search

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  • $\begingroup$ Great analysis, and the only one including Power. Re Mars, g is lower, but Mars is also smaller, so less need for high speed trains. $\endgroup$
    – Roland
    May 8, 2023 at 13:06
  • $\begingroup$ @Roland "Close enough" circumference of Terra and Mars are 40,000 and 20,000 km. Once there is enough population centres (say Kim Stanley Robinson Mars trilogy scenarios) then settlements will still be very widespread. Distances of 1000 km would be liable to be common between clusters of population and end to end trips of 5000-10,000 km highly desirable - if not always practical. It's likely that IF rail could be fast and energy economic that rail could be the predominant transport mode. "Air"craft may be viable with eg thermal transfer engines but lack of atmospheric oxidant will ... $\endgroup$ May 9, 2023 at 9:14
  • $\begingroup$ .. make that hard. SO an eg 1000 km/h train could be immensely attractive. Martian atmospheric density is about 0.02 kg/m^3 (as opposed to 1.2 at earth sea level so power at 1000 km/h (280 m/s) will be about Cd x 280^3 x 0.5 = 220 kW per square metre of frontal area at Cd of 1. Assume Say Cd=0.3 and A of 10 m^2 giving power of 660 kW - say 1000 kW actual. This is for a "not too long " train - friction and body drag add with length. [See by.genie.uottawa.ca/~mcg3341/AerodynamicsOfHighSpeedTrains.pdf - their use of Cd is unfamiliar to me. I'll try to resolve. My figure is useful as used. $\endgroup$ May 9, 2023 at 9:29
  • $\begingroup$ @Roland See my addition to answer. I disagree with you :-) - but your point was useful. $\endgroup$ May 9, 2023 at 9:39
  • $\begingroup$ Great additions! But let me throw in one little pesky problem. Energy source. Less atmosphere means less resistance and less power needed, but where to get energy from. No previous life on Mars means no fossil energy. Far from Sun plus year long dust storms = less solar power. Only 1% atmosphere = no wind power, no oxygen. No water = no hydro power. No uranium ore, no steel industry, or mining and reactor building needs heavy materials and components to be flown in from Earth which is almost impossible. $\endgroup$
    – Roland
    May 9, 2023 at 14:23

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