Field
Spaceflight
Celestial mechanics used forwards — where to burn, and what it costs.
The cheapest way between two orbits, and why it is so slow
Two burns and a long coast is the least fuel that will move a spacecraft between two circular orbits. It is also, for anything beyond the Moon, an unreasonably long wait.
The speed that does not come back, and the √2 that separates it
Escape speed is exactly the square root of two times circular speed, at every distance from every body. Being in orbit is already 71% of the way to leaving.
Stealing speed from a planet, which does not notice
A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.