The don’t call it “The Tyranny of the Rocket Equation” for nothing.

More payload means more fuel. More fuel itself means more fuel.

That looks like an argument that you need more than twice as much fuel to get twice as much payload to orbit -- that fuel requirements go up more than linearly.

Maybe this is obvious, but two rockets can get twice as much payload to orbit using twice as much fuel. You can that up linearly all day long. (It holds for "twice as much", but the same argument doesn't work for "twice as fast" or other metrics.)

X fuel can get Y payload to Z speed. To get that payload to 2 * Z speed, you need 2 * X + (X * (X / Y)) fuel, because the fuel counts as part of your payload for the first Z of speed.

If X is 500t and Y is 5t, you now need 51,000t of fuel to achieve 2 * Z speed.

To augment what Duskstar said, "twice as fast" is what counts: to get something to orbit with a steeper gravity well, you need more of a velocity change.

I highly recommend Kerbal Space Program for anyone interested in getting a feel for orbital mechanics. Please buy this game for any young people in your life.

It’s a great game, that will ruin your appreciation for basically all space movies.

It also made me enjoy seveneves

It's also worth mentioning that once you are in space things are relatively easy. You only need around 20% more deltaV to get from Earth to Mars, as from Earth to the Moon (assuming a one way trip).

Would it be possible for the numbers to be a bit more difficult to work with so maybe we could go to low earth orbit, but never be able to make it to the moon?

Once you are in orbit, anything else is possible. Because if you can get something small to orbit, you can get a lot of small things (such as additional fuel) to orbit, to reach escape velocity. That is because once in orbit you no longer have a need to reach the next velocity stage as rapidly as possible, which is why ion engines are so efficient but can only be used after LEO.

If you can make it to orbit, you can make it anywhere (it just may take multiple launches to assemble/fuel the spacecraft going elsewhere).

Wait, really? So what's the deal of first and second cosmic velocity?

I guess the idea is that if you can make it to orbit (even if you waste all of your fuel), then you could in principle refuel in-orbit (by getting another unmanned rocket with just fuel into orbit) and thus escape the orbit.

But yes, you still require more energy (exactly sqrt(2) times as much) to "escape" Earth's gravity entirely than to orbit it with a circular orbit.

Exactly twice as much energy. √2 as much velocity.

You're quite right -- that was a typo. ;)

Making orbit is a single hard step; raising orbit, which is all that making it to the moon requires, can be done more gradually (with low-thrust/high-efficiency engines, on-orbit resupply, etc).

So if you can get off the planet you can get anywhere... but that doesn't mean cost-effectively. If your planet is such that it takes a Saturn V to launch a cubesat, it may simply cost too much to make it to the Moon.