[bruce511]

Even if the math of the arc length was correct (and you don't need to be a math professor to figure out it isn't) there's another logic misstep.

Implied in the caption is that the speed is the same at all heights (given that an increase in distance is implied as an increase in time.)

This is again obvious nonsense - speed is a function of thrust versus drag, and it's safe to say that both of those are affected by air density.

It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down.

[mastermedo]

> It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down.

Can you expand on this? My brain is not connecting the dots.

[db48x]

He is talking about orbital mechanics, rather than free space. When you are in an orbit, the shape of the orbit is determined by your speed. At every distance from the center of the object you are orbiting (such as the Earth), there is a speed that makes your orbit a circle. If you are going at any other speed then your orbit will be an ellipse instead. Too fast and your orbit rises higher above the Earth. Too slow and it dips back down closer to it. If you try to “catch up” with an object ahead of you in your orbit by speeding up you will only turn your orbit into an ellipse that gets further away from the Earth, and thus further away from the object you were trying to catch. Instead of catching it you’ll go up and over it. As Niven wrote, “forward is up, up is back, back is down, and down is forward”. It’s rather counterintuitive at first. Playing KSP can help you get a feel for it, especially once you start docking multiple craft together.

[Sharlin]

It’s even worse than that. By speeding up you end up actually getting further behind your target because in your new higher orbit you actually move slower on average, and as your average orbital radius gets longer, so does the circumference, so you end up on a "detour" trajectory compared to your target!

Whereas if you slow down, you drop to a lower, shorter, higher-speed orbit.

[davrosthedalek]

Just to point out here what's different between "space" and "not space": "Space" assumes no "height control",i.e. ways to exert force "down or up" along the earth-object direction. That's obviously not true for a plane. If you can exert force in that direction, you can change speed and keep the shape of the trajectory around earth constant.

[falcor84]

Wait, why wouldn't an orbiting satellite not be able to apply its thrusters "down"?

[davrosthedalek]

They could. The point is that if you allow for that (and it they would need to be constant on), the more speed -> either higher orbit or more elliptic orbit doesn't hold. But no satellite has enough fuel to keep its thrusters on indefinitely.

[nyc111]

This is called Kepler's Third Law, right? Radius^1.5 :: Period

[f1shy]

You are literally going way and beyond what the target audience of that post (the original with bad math) was for.