There's something backwards about orbital mechanics that always tripped me up. Outer planets have longer years than inner planets, and higher satellite take longer to orbit than lower ones. That's not just because the orbit is a longer path, the orbital speed is literally slower in higher orbits. The general formula for orbital speed is clear: as r increases, v decreases.

Say you're in a circular orbit near Earth with a constant speed of 10. Call this level "Low Orbit", and you want a higher, slower orbit at a level called "High Orbit". But you can't just slow down - that will send you tumbling down toward Earth.

Then to reach a higher orbit, with a lower speed, you'd have to... speed up? How does that work?

You fire your engines for a short burst, boosting your speed from 10 up to 12. This sends you climbing higher, out of your circular orbit, like a car coasting uphill.

You are in a new elliptical orbit. This orbit speed is no longer constant, it drops steadily as you coast uphill - trading speed for height. By the time you reach the highest point of your elliptical orbit, you've slowed to 6, less than your original speed. You cleverly boosted just enough that the crest of the hill is at "High Orbit" level. This elliptical orbit is called a Hohmann Transfer orbit. Its low point is your old orbit - right where you fired your engines, and its high point is at "High Orbit".

And like a car coasting uphill, you will start falling back down hill if you do nothing else.

To stay at this higher altitude, fire your engines again. Checking the formula above, you see the circular orbital speed at this height is 8. You accelerate from 6 to 8, and your speed now stays at a constant 8. You left your elliptical transfer orbit and started a new circularized orbit.

Orbit Shape	Speed	Height
Circular	constant 10	Low
	Accelerate from 10 → 12
Elliptical	between 6 & 12	between Low & High
	Accelerate from 6 → 8
Circular	constant 8	High

You started with speed of 10. You accelerated twice, and ended up slower. The trick was to travel uphill and let your speed fall in between. You added kinetic energy but converted that into gravitational potential energy, in a higher circular orbit, thanks to the beautiful & backwards effects of orbits.