Can you see a coin from 400 kilometers?

Hi,
no, probably not with naked eye, I think, though if it would be some special coin, maybe? The point is that you can use some cool things to see such a coin, for example Hubble Space Telescope!

Ok, I found some page with physics problems and one of the first was to calculate what is the smallest angle that Hubble Telescope can distinquish. I calculated that it is roughly:

0.0127”

This means 0.0127 arcsecond!!! How far away football has to be to have the angular diameter of one arcsecond

What about a human eye how good is it? Well eye can distinquish only things that are 1′ away from each other , which means one arcminutes, one degree has 60 of those so it is pretty good but not so much as Hubble. But anyway, back to the title, how small is a coin?

I measured the second largest Czech crown to have 2.55 centimeters.

Now how far away do you need to be to not be able to see it with naked eye? We will use this formula to get the diameter:

θ=D/r

Where the diameter (D) is 0.0255 meters. The distance (r) is what we are trying to find and the angle eye can distinquish is θ (in radians). 

r=D/θ

If all the calculations went right it should be:

r=87.6625 meters

But beware this is not counting air, humidness and so on so you will most certainly not be able to see coin on this distance.

What about the Hubble telescope? We can use the same equation but for θ we will insert much smaller value!

And yes, you can see the coin from 414,153.744827 meters! Which means that you can see one freaking coin from London 50 kilometers behind Paris! One coin!

This is so cool.

Dragallur

You can try this by drawing two dots on paper and then moving away from it, at one point when you are far enough you should be able to see only one.

This equation is simplified but it should work for small angles.

Again, this works only in space because there is nothing that would block your sight.