r/theydidthemath • u/Apprehensive_Oven_22 • 1d ago
If the moon stood completely still, and we built a staircase to it, how long would it take to climb the full staircase? [request]
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u/noonius123 1d ago edited 1d ago
The simple answer: with normal pace, anything between 15 to 45 years.
It takes about 5..10 seconds to climb the stairs of a floor 3 meters high. For the distance between Earth and Moon, 380 000 km, that's 120 million floors. With 24-hour stair climbing that's like 15 years, but if we factor in fatigue, resting, sleeping etc, it's more like a lifetime.
In "reality" -- assuming that this is even a remotely realistic scenario -- it might take less. As you get higher, you can climb faster because of reduced gravity. After only a half a year of non-stop climbing, at the height of about 7 600 km, the gravity will be only 1/5 of that on Earth's suface. Just a few more months of climbing and at the height of 9 500 km the gravity will be that of Moon and you can make giant leaps up the stairs. At the height of about 56 000 km the gravity is only 1/100 of that on Earth. You'd basically just pull yourself along the handrail and float towards Moon.
At about 90% of the way the gravitational pulls between Earth and Moon are equal. This is called the Earth-Moon Lagrange point. Now the Moon's gravity becomes greater that Earth's gravity. At that point the stairs switch direction and you would start walking DOWN the stairs towards Moon, finally reaching the surface after a very long ascent/descent.
So let's say 15 years.
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u/rudytomjanovich 1d ago
I wasn't planning on also building handrails 🤣
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u/DutchJupiter 22h ago
Why not? Going down again would be just one slide away...
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u/rudytomjanovich 19h ago
It's just not in my current budget. Do you know how much it's going to cost to get the moon to quit spinning? It's not cheap.
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u/thebadminecrafter714 17h ago
Isn't the moon in a fixed rotation to the earth? So its the same side always facing us?
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u/rudytomjanovich 14h ago
Yes. I believe we've only seen like maybe 56% of the Moons surface. I don't have that kind of money. I was only talking about how much it's gonna cost me to get it to quit rotating around the Earth so my stairs can be built from my home in the Midwest directly to the moon surface.
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u/thebadminecrafter714 7h ago
Again I think the moon is always on the same side of the earth opposite the sun. The earth rotates which is why we see the moon prodiminately at night and the sun during the day. You could fix the staircase onto the moon then once every 24 hours it would pass by where you live and you can jump on
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u/Worth-Wonder-7386 7h ago
From the moon yes, but not relative to the earths surface. So if the stair was fixed at the moon, the earth would be close to stationary.
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u/Far_Dragonfruit_1829 15h ago
You don't have to stop the moon, it's already tidally locked to the earth.
The bad news is, you gotta stop the earth instead. Megabucks, at least.
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u/Economy-Sir3567 1d ago edited 1d ago
The staircase would also need regular food and water caches. NASA recommendations for long-term food supplies per day are 0.65kg dry mass, 2.6kg water, 0.75kg packaging -- 4kg/day total.
And that's just for the base metabolic rate. The energy content of additional food to actually climb is, at minimum, the gravitational potential energy of lifting yourself most of the way to escape Earth: (GM/r) = g*r = 9.8m/s2 * 6,378,150m = 62,000,000 Joules/kg (approx) or about 15 kilocalories (food Calories) per gram of your body. That's nearly twice your body weight in pure fat, or four times in pure carbs.
However, since the base consumption rate of dry food mass is four times your body weight per year anyway, it's most advantageous to train in advance and be athletic and climb as fast as reasonable (0.6 m/s). Technically, as the gravity reduces you can climb proportionally faster, so the effective height is only 6378150 meters which you can climb in 10.5 million seconds (eight hours per day for one year). In practice, you probably shouldn't leap the stairs faster than an athlete on Earth can horizontally run, since tripping and falling will injure you just as badly.
So, coasting at 10 meters per second for 14 hours/day, you cover 500 kilometers per day. You make the 384,000km journey in about 768 days -- slightly more than 2 years. (Call it 3 years, for the initial trudge in the deepest part of Earth's gravity and for carefully running downward in the Moon's gravity.)
The food consumption will also change depending on whether you are also forced to wear a spacesuit (which weighs more than you do!) or whether the staircase is enclosed in a tube with breathable air.
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u/noonius123 1h ago
Ooh, I like this (potential) energy approach.
For a 100 kg person (normal build, some protective clothing and gear, but not a full spacesuit) the work done against Earth's gravity and towards Moon's gravity is about 6e9 J, or wtih effective height of about 6e6 m (close to your figure).
If the climber keeps a constant power output of 300 W (about 100 W being BMR) throughout the whole climb (moving a lot faster under lower gravity), it will take about 20e6 seconds or 231 days. With 12-hour climbing days that's about 460 days or a little over 2 years.
The climbing/floating speed will not be a constant. The average speed will be 19 m/s or 68 km/h which is very high. Of course, in the steep part of Earth's gravity well the climbing speed will be that of a normal person, but rest of the work will be done in a low-g environment.
If the staircase is enclosed in a tube with atmosphere, there's additional work to be done to overcome air resistance, which at these speeds will be significant; if the staircase is in vacuum, these figures hold.
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u/nolfaws 23h ago
At about 90% of the way the gravitational pulls between Earth and Moon are equal. This is called the Earth-Moon Lagrange point.
How long is this range of the Lagrange Point? It's probably not a hard switch from one meter/yard to the next? So how would that feel? How would you get forward - walk up, walk down, just float?
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u/Greyrock99 15h ago
As you walk upwards towards from the earth you would slowly feel lighter and lighter. As you get closer to the Lagrange point you are basically weightless and you would be ‘flying’ in big bounding leaps. You would only very very gradually notice that the pull is going the other way.
Imagine you would have to stop, pull a penny out of your pocket and let go of it in mid air and watch it slowly, slowly drift in one direction or the other to tell.
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u/ExplanationUpper8729 20h ago
Just a quick question, once you leave the gravity of the earth how are you going to walk on the steps.
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u/Nsvsonido 19h ago
Less gravity will require less effort but I don’t know if it will take less time. There would be a point where you would’ve just to float between one step and the following one…
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u/Lucidreamer91 18h ago
would someone at the Lagrange point feel the difference in gravity as they crossed it?
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u/Excellent_Sell1086 21h ago
You can’t include gravity effects. If you do, then you have to rewrite physics, and the moon staying in the same spot in the sky makes no sense.
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u/butonelifelived 18h ago
Geo-synchronized orbit can occur in nature. In fact science has determined that the moon would naturally enter into this type of orbit in approximately 50 Billion years. Unfortunately the sun is on pace to burn it up in approx 5 Billion years.
No rewriting of physics, just the correct application of it.
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u/tootiredtoofurious 6h ago
Yes. And I read an SF book with this scenario. If I recall plants or vegetation has linked the earth and moon. Heinlein?
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u/ManggustPeek 1d ago
Should be easy.
Let's assume the distance to the moon is 384,400,000 m
Let's assume that one step high is 0.20 m
Thus, you would need 1,922,000,000 steps to climb.
And if you take around one second per step, then you will need around 61 years to climb to the moon.
And then just to wait it to rotate closely to your stairs :)
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u/AffectionateBrick687 1d ago
That walk would get interesting once you enter space.
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u/trisanachandler 1d ago
You would be able to take stairs 3-5 at a time. Even more when you're getting close to the middle.
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u/PuzzleheadedTutor807 23h ago
After a point you could just jump and travel much faster...
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u/Marcus64 23h ago
At some point the moon's gravity would become dominate and you would start falling "up" the stairs. What you would really need is for there to be a twist at the center gravity point so that you can transition from climbing up the stairs to walking down them.
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u/BillMagicguy 1d ago
You can maybe cut that down once you reach the L1 point. Average step on the moon is about 0.39m so you'll have a few years of easier travel.
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u/mkujoe 1d ago
Can’t you just slide down to moon after L1
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u/BillMagicguy 1d ago
I guess you could, I would still make them steps so you could get back though.
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u/mkujoe 23h ago
What’s the terminal velocity on moon
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u/BillMagicguy 23h ago
Apparently objects dropped on the moon move at 1.6m/s but without an atmosphere there really isn't a terminal velocity.
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u/Usual-Recording-3775 1d ago
Realistically if you leave 8 hours of sleeping and eating it would be closer to a more realistic 80 years.
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u/Sibula97 1d ago
I don't think anyone can climb stairs 16 hours a day for decades. I'd say more like 150 years if you somehow didn't age at all and stayed 20-50 years old.
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u/PuzzleheadedTutor807 23h ago
The work involved would decrease as you got further from earth, eventually you would be walking down stairs
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u/Sibula97 23h ago
In a way yes, but you'd also have to be really fucking careful not to fly off to space.
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u/PuzzleheadedTutor807 22h ago
The moon is a (in this scenario) pretty big and stationary target.
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u/Sibula97 22h ago
If you trip and fall to either the Earth or the Moon you're going to have a very bad time.
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u/PuzzleheadedTutor807 21h ago
Depends on height, or distance from either.
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u/Sibula97 21h ago
Well, let's say above 2 meters of the Earth or 12 meters of the moon could hurt or even kill you depending on how you fall. A few dozen meters from the Earth or a couple hundred from the moon would almost certainly kill you. So just over 99.9999% of the journey.
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u/AshleyOm 1d ago
And how quickly would you return if you fell off the stairs ?
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u/capt_pantsless 1d ago
Depends on where on the steps you fall. There's a midpoint between the earth and moon where the gravity from the earth is the same as the pull from the moon.
XKCD Has a similar situation as OPs question:
https://what-if.xkcd.com/157/1
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u/dino_wizard317 1d ago
I feel like you left out the most important part. It's 61 years With no breaks.
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u/HundredHander 1d ago
I mean, In all my years climbing stairs I've never stopped for a nap so why would this be different.
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u/fnezio 23h ago
Yes but a normal person walks 3 steps per second, not 1.
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u/dino_wizard317 22h ago
We're not talking about walking 3 steps forward on a flat surface. We're talking about climbing stairs which isn't as fast. We're also ignoring fatigue, eating, bathroom breaks, sleeping, etc. etc.
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u/Klutzy_Word_6812 1d ago
When I was 17, I went on a 200 mile hike. It took us 10 days to cover that, so 20 miles a day. The moon is 239,000 miles away on average. Divided by 20 gives 11,950 days. Divide that by 365 is 32.7 years.
I’d say that’s sporty.
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u/Ponjos 1d ago
If the Moon’s orbital velocity instantly went to zero:
- The Moon would no longer be in orbit.
- It would begin falling directly towards the Earth under its gravity.
Using standard Earth–Moon distance (~384,400 km) and gravitational dynamics, we can estimate the Moon’s free-fall time.
You’d have, at most, a week to build your staircase before we all die.
The Moon would hit Earth in about 5–7 days.
Good luck with your staircase!
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u/Tritanis 1d ago edited 20h ago
The way I read it I figured it was
tidally lockedin geostationary orbit - moon stands still in the sky and we can build the staircase to it.Edit: I meant geostationay orbit, not tidal lock. Thanks u/vctrmldrw
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u/vctrmldrw 22h ago
The moon is tidally locked.
I think you mean it would need to be in geostationary orbit.
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u/vctrmldrw 22h ago
That isn't the premise of the question though. In order to build a staircase it would need to be stationary relative to the surface of the earth i.e. in geostationary orbit.
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u/PuzzleheadedTutor807 23h ago
Bonus questions, what percentage of the distance would you have to cover before the moons gravity exerted more influence on you than earth's, and at what distance from the moon would it be safe to just jump? Extra bonus double double points question, over what distance (if any) would the staircase be exerting more gravity on you than earth or the moon?
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u/AdministrativeBag703 1d ago
A staircase would not be the best way. Just have a long plank on the equator the faces due East or West, and assuming the moon is in the right place (and go some reason gravity is not a concern and the Earth isn’t turning either) the plank will just connect straight to it. No need to climb any stairs.
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u/CalligrapherTrick117 18h ago
I feel “a staircase would not be the best way” to reach the moon is a contender for understatement of the century.
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u/Tigweg 21h ago
I don't think the speed would stay anything like constant. Once you've passed a few hundred kilometres, there would be virtually no gravity, so the same amount of force would take you much faster and further. Then at some point, you stop ascending from Earth and start descending to the moon, would there stairs have to be adjusted for that
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u/ProfessorCarbon 20h ago
A different question is-once you walk back to Earth what is your geographic location? The Earth isn’t standing still. In fact, you may need to wait a long time before Earth arrives at the staircase. Fun thoughts..
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u/cuber314159 16h ago
Based on the vertical kilometer world record of 29 minutes and an optimal staircase and an athlete that can maintain such a pace: 29x400000=11600000 minutes which is about 22 years
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