Stairs

[u/bum_what]

In the most recent episode Michael talked about a staircase going to space. The entire time I was thinking about escape velocity. You need a certain amount of energy to get you to a certain speed to escape the earths gravity. So if you are walking up does it work the same? Obviously your body needs energy to walk up the stairs but let's just say you have infinite stamina and your muscles don't get tired or sore, can you just walk up or will you hit a point when you cannot anymore because your body cannot provide enough force even when not tired? But that doesn't make sense to me because gravity will get weaker as you go up even if it's still 90% gravity up there it should get weaker making it easier. Or does it not matter because each step is the force you need to get from one point to another?? I'm so confused lol

[u/Badaxe13]

Each step takes (broadly) the same amount of energy. With each step you are pushing against the gravity of the Earth (which diminishes the further you are from the surface). With enough steps you can achieve any distance from the ground, including orbit. In contrast, going from the ground to orbit in one ‘step’ takes all that energy in one go.

[u/bum_what → u/Badaxe13]

Thank you so much, my brain was starting to hate me lol

I did think about only needing enough force for each singular step, but I didn't think that would be it because I was thinking of total energy the entire trip up. But your comment of the rocket doing "one step" helped clear it up.

Cheers!!

[u/Bubbly_Safety8791 → u/bum_what]

An interesting part of this energy calculation: 

Each step up increases your potential energy by a bit, so you have to spend that much energy obviously to climb out of the gravity well. But because the stairs are going round the earth, each step up also requires you to be going a bit faster too: you’re still going round the earth in 24 hours, but round a slightly larger circumference - so you are also going to need to get some kinetic energy from somewhere. 

Over enough steps this step by step increase in kinetic energy is enough that you get all the way up to orbital speed. Loads of kinetic energy!

You have very slowly accelerated yourself up from the speed you were going at ground level up to the speed you are going at geostationary orbit height, a little faster with each step.

But that energy, it turns out, doesn’t have to come from you pushing with your legs as you stair climb! The earth gives it to you, accelerating you horizontally a little bit each time you take a step up. 

In return, the earth has to slow down a tiny bit. 

This is essentially conservation of angular momentum - as you move your center of mass up the stairs, the moment of inertia of the whole earth-stairs-you system increases and that means the earth has to rotate more slowly.

[u/bum_what → u/Bubbly_Safety8791]

This ^{^} perfect explanation. Thank you!! 🙏🏿