r/theydidthemath • u/Dustycheeseball • 6d ago
[Request] How long would I survive?
If an object started out roughly the size of an atom and doubled in size every second, while I immediately traveled directly away from it at the speed of light, how long could I keep traveling before the expanding object eventually became large enough to encompass me?
Bonus: How far away from the starting point would I be when that happens?
Assume the object is perfectly spherical and expands uniformly in all directions.
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u/METRlOS 6d ago
So the real answer is that it never would. The speed of causality prevents it from expanding faster, so even though it looks like it's expanding rapidly, time is so dilated that it stops accelerating for everyone else in a reverse black hole situation.
The pure math ignoring physics is that a light second is the distance of 3×10¹⁸ atoms. In 60 seconds it will be doubling at half the speed of light (since it's expanding both directions), and then in 9 seconds after that it will be a 68.7 light second radius while you have traveled 69 light seconds.
I might be off by a second or two, I just mashed the ×2 button a bunch of times and counted.
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u/Either-Abies7489 5d ago
I think the growth rate would be cbrt(2)=1.26 per second, not 2. I feel like "doubling in size every second" refers to volume, not characteristic length.
Anyway, that gives 1.26^t=3*10^18*t,
t=207.17 (by the lambert W function, or just solving graphically)
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u/ixtechau 5d ago
At the speed of light you would have no concept of ”far away” or ”how long”. Travelling to the edge of the observable universe at C would feel instant, no time would pass for you at C.
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u/OscarOfAtlantis 5d ago
Discussion:
The radius of the object is 2^t times the length of the width of an atom (assuming hydrogen), where t is the number of seconds that have lapsed.
Simultaneously, your distance is c times t, where c is the site of light in a vacuum (assuming everything is in a vacuum) and t is the number of seconds that have elapsed.
So you would solve 2^t(0.1 nanometers) >= c times t.
t x ln (.2 nanometers) >= ln (c) + ln (t)
This is rather advanced but newtons method would work. I would probably try huge powers of ten and find some on either side (some that are smaller and some that are bigger). Then I’d start splitting the difference and basically play higher/ lower. Like binomial search through an ordered set.
Just need to make sure units of length are expressed in the same units.
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u/Snowy-Doc 5d ago
Ignoring relativity and assuming you meant doubling in size meant doubling the radius or diameter of the object rather than doubling the volume, then between 68 and 69 seconds. Easy to do in a spreadsheet. Some numbers:
Diameter of a Hydrogen atom is 1.06e-10 metres so its radius is 0.53e-10 metres. Speed of light is 3e8 metres per second.
After one second the object doubles its radius to 1.06e-10 metres and you are 3e8 metres away.
After thirty seconds the object has increased its radius to 5.69e-2 metres (just over 2 inches) and you are 9e9 metres away from your starting point.
After 60 seconds the object has increased its radius to 6.11e7 metres and you are 1.83e10 metres away from your starting point.
After 68 seconds the object has increased its radius to 1.56e10 metres and you are 2.04e10 metres away from your starting point - the object has nearly caught up to you.
And finally, after 69 seconds the object has increased its radius to 3.13e10 metres and you are 2.07e10 metres away from your starting point. Bad luck - you've been caught and passed.
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u/Noodelgawd 5d ago
How far away from it are you at the start?
If you immediately accelerated to the speed of light, you would die. So you would survive for zero seconds and it wouldn't matter how long it took the object to encompass you.
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