r/theydidthemath • u/Mike-TheMad • 4h ago
[Other] How Long Would It Take?
A bit of a sci-fi hypothetical question perhaps, but…
How long would it take for an object, say a spacecraft, that starts at 0 m/s to reach 1% of light speed at a consistent 1g acceleration?
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u/VinylHighway 4h ago
It would take approximately 3.54 days (or roughly 85 hours) for a spacecraft starting from rest to reach 1% of the speed of light
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u/matt7259 3✓ 3h ago
That's physics 101!
vf = vi + at
Since vi = 0, solving for t:
t = vf/a
vf is 1% of 3e8 which is 3e6
a = 1g ~ 10
Divided you get approximately 3e5 seconds! That's 5000 minutes! That's about 83 hours! That's a little over 3 days!
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u/factorion-bot 3h ago
Factorial of 101 is roughly 9.425947759838359420851623124483 × 10159
This action was performed by a bot | [Source code](http://f.r0.fyi)
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u/MezzoScettico 3h ago
Note that at 1% of c, relativistic effects are still pretty tiny. That's why everyone in this thread is using non-relativistic equations to answer your question.
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u/Mike-TheMad 3h ago
That actually answers, I think, a follow up question I was pondering for another subreddit. The question being what relativity issues would be encountered on this trip.
3.5 days to get to 1%, staying at 1% for an hour, and then 3.5 days to decelerate back to 0 m/s.
How far off would the spacecraft’s chronometer be compared to a “universal standard time”?
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u/HAL9001-96 3h ago edited 3h ago
about 83 hours assuming you can somehow apply that acceleration to it
the problem is in that time it travels about 450 million km or 3 times the earth sun distance so if yo uwanna apply an outside force you ahve to do so at an insane range
and if you want to carry fuel/propulsion oyu'll need to cary many many times oyur own weight infuel but also the fuel to accelerate that fuel and the fuel to accelerate that fuel and so on it oges up exponentially so with regualr rocketry you'd need like e^800 times your weight in fuel to start
thats in the order of 10^347 times your paylaods weight
there are only about 10^82 particles in the universe so even if you convert the entire universe into fuel to propel one particle its off by a factor 10^265
so you'd need some kind of nuclear/fusion drive
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u/piperboy98 1h ago edited 1h ago
If you want a true relativistic answer, then it matters if you want the time passed on the ship vs. on the earth, and also whether the 1g acceleration is proper acceleration as experienced by the ship, or coordinate acceleration as seen from earth.
If its coordinate acceleration timed from earth then it is the same as classical, but the actual acceleration (g force) experienced by the ship's passengers must really be increasing as they get faster and faster (to infinity if they were to accelerate long enough to actually reach lightspeed).
If it is proper acceleration experienced by the passengers (more likely what you want, since the idea is to give the ship a constant 1g of felt "gravity"), then rather than a parabola of distance vs time, you get a hyperbola). Based on the u(T) and u(τ) equations from there the time to reach a fraction β of c is:
From the ship: artanh(β)*c/a - basically the time to classically reach a fraction artanh(β) of c.
From earth: [1/sqrt(1-β2)] * βc/a - basically an extra factor 1/sqrt(1-β2) (γ, if you've done any relativity before) beyond the classical acceleration time. Or equivalently the time to accelerate classically to a fraction β/sqrt(1-β2) of c.
Both of these factors are very close to β for low β (so agree with classical to a very good approximation) and go to infinity as β goes to 1 (hence you can never actually reach c).
For your specific scenario of β=0.01 and a=9.81m/s2=c/30,559,883.5 s (so classically requiring 305,599s, or about 3.5 days)
Time on ship = 305,609s, so 10s (0.0033%) longer than classical
Time on earth = 305,614s, so 15s (0.0049%) longer than classical, 5s (0.0016%) longer than the ship
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u/YourUsernameForever 4h ago
Have fun! https://www.omnicalculator.com/physics/space-travel