Consider the astronauts in the International Space Station (ISS), which is a giant spaceship orbiting earth. They aren’t walking around with their feet on the floor like you or me on earth - they’re floating. Why? Why are they floating and we aren’t?

As usual, I recommend trying to think about this for yourself before reading on; the explanation will be much more interesting that way.

I’ve asked a few people this question and so, instead of going straight to the answer, let’s follow a common train of thought.

Maybe it’s because there isn’t any gravity up there? Ok, maybe not no gravity, but very little. It’s like when Neil Armstrong took a step on the moon and seemed to float for a while before coming back down to the moon’s surface.

Good thought, but no. The earth’s radius is about 4,000 miles. The ISS is only about 250 miles above the earth’s surface. That’s nothing! If you took a huge step back and looked at the earth from afar, you’d barely be able to tell the difference between us on the earth’s surface and the astronauts in the ISS.

If you want to get numerical, here’s the formula for gravity:

$F = G \frac{m_1 m_2}{r^2}$

So how much less gravity do you feel on the ISS than on the earth’s surface? Well, the only difference is $$r$$. $$4000^2/4250^2$$ is about 0.88. So, the force of gravity is noticeably less on the ISS, but only by about 12%. That doesn’t account for the difference between walking on the ground and floating through space.

Does it have something to do with the fact that they’re orbiting? Like, when you’re in orbit, gravity is pulling you towards the earth with just the right acceleration (relative to your tangential velocity) to keep you at exactly the same distance from earth. Am I on the right track?

Definitely on the right track, although here’s a hint. Even if they weren’t in orbit, I still think they’d be floating. Let’s say their tangential velocity wasn’t fast enough to keep them perfectly in orbit, so instead they were spiraling in towards earth, I still think they’d be floating. Or go the other way - say their tangential velocity was too fast and they were spiraling away from earth. I still think they’re be floating.

Oh… it’s like skydiving then? You’re falling? But if feels like floating?

Yes, exactly! The astronauts are in “free fall”, meaning that the only force acting upon them is gravity. Why doesn’t the space station exert any force on them? Because the space station is also in free fall, and it’s falling at exactly the same rate!

Something important to remember is that gravity will accelerate two objects at the same rate, regardless of their mass. This is because gravity exerts a force proportional to mass, but - for a given force - acceleration is inversely proportional to mass. Those two effects end up canceling out so that acceleration due to gravity does not depend on mass. Probably easier to see it with symbols than with words:

\begin{align} F &= G \frac{m_1 m_{earth}}{r^2} \\ F &= m_1 a \\ G \frac{m_1 m_{earth}}{r^2} &= m_1 a \\ G \frac{m_{earth}}{r^2} &= a \end{align}

So, acceleration (of an object) depends on the mass of the earth, but not the mass of the object.

Where were we? Right, so both the astronaut and the spaceship and everything else around them are falling towards the earth at exactly the same rate, which means they don’t exert any forces on each other.

Here’s an analogy: What if you and a friend jumped off a cliff together? You’d both “float” right next to each other (while careening to your doom). Air resistance would make that example feel less serene than it does for astronauts, but the analogy mostly holds. Let’s say that instead of jumping, you and your friend were in the back of a large van (not strapped in) as it drove off the cliff. Then you both would be floating (inside the van), much like two astronauts inside the space station.

Cool! But if being in orbit isn’t required, does that mean that astronauts are always floating?

Yes, anytime the spaceship that they’re on doesn’t have its engines turned on. If the spaceship turns on its engines, that will accelerate the spaceship. Assuming you were previously floating inside it, eventually a spaceship wall will “bump into you” and exert a force on you to keep your acceleration in line with the ship’s.