How about a quick little circular motion exercise, since that’s what I’m teaching in my recitation at the moment? We know the force equals mass time acceleration, so how about we put the force of gravity on the right side and the force required for uniform circular motion on the left:

And we’ll solve that for the velocity:

That’s the velocity you need to reach in order to attain circular orbit at a distance r above the earth’s center. M is the mass of the Earth (or whatever you happen to be orbiting, and m is the mass of the thing in orbit. Of course m cancels anyway. But what if we find the escape velocity by setting the initial kinetic energy equal to the potential energy at the surface of the planet? We get this:

Solve for v again:

Looks a lot like the expression for orbital velocity! We can compare them directly by dividing. Let me label orbital velocity with “o” and escape velocity with “e”:

So no matter your mass, or the mass of the planet, or the escape velocity, or any other detail the escape velocity is always 1.41 times greater than the circular orbit velocity. Tremendously important? Nah. But a cool problem nonetheless.