# Anyone remember their calculus? Time to get to Alpha Centauri? **URL:** \ **Category:** Factual Questions\ **Created:** [December 31, 2011, 7:16pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967 "2011-12-31T19:16:08Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![CanTak3](https://avatars.discourse-cdn.com/v4/letter/c/f14d63/32.png) [@CanTak3](https://boards.straightdope.com/u/CanTak3)\ **Post date:** [December 31, 2011, 7:16pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/1 "2011-12-31T19:16:08Z")
Assume 4.37 light years away. Obviously the fastest possible trip would be for light and would take 8.74 years round trip. But I’m more curious on two things: - How much time passes on earth while the human (below) makes the trip? - How much time passes for that human since they are travelling so fast? So, far in the future, mankind has a spaceship with an unprecedented amount of power. The spaceship accelerates at the earths gravitational rate. It does this until it reaches the halfway mark, then the ship rotates 180º and it accelerates in the opposite direction at earths gravitational rate. When it makes a stop, it’s at Alpha Centauri (or it’s close enough!). ---
**Author:** ![obfusciatrist](https://avatars.discourse-cdn.com/v4/letter/o/4af34b/32.png) [@obfusciatrist](https://boards.straightdope.com/u/obfusciatrist)\ **Post date:** [December 31, 2011, 7:28pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/2 "2011-12-31T19:28:35Z")
> [@CanTak3](#): > > Assume 4.37 light years away. Obviously the fastest possible trip would be for light and would take 8.74 years round trip. But I’m more curious on two things: > > - How much time passes on earth while the human (below) makes the trip? > - How much time passes for that human since they are travelling so fast? > > So, far in the future, mankind has a spaceship with an unprecedented amount of power. The spaceship accelerates at the earths gravitational rate. It does this until it reaches the halfway mark, then the ship rotates 180º and it accelerates in the opposite direction at earths gravitational rate. When it makes a stop, it’s at Alpha Centauri (or it’s close enough!). The answer to the first one is 8.74 years. The answer to the second, if they are truly moving at the speed of light, no time will pass at all (but they wouldn’t be able to do that). Since they aren’t moving at the speed of light, since they can’t, it would be necessary to know how close they are to doing so. Not sure what you’re asking for in the next part (how long the trip will take? How fast they’ll end up going?) ---
**Author:** ![obfusciatrist](https://avatars.discourse-cdn.com/v4/letter/o/4af34b/32.png) [@obfusciatrist](https://boards.straightdope.com/u/obfusciatrist)\ **Post date:** [December 31, 2011, 7:37pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/3 "2011-12-31T19:37:07Z")
The formula for figuring out time dilation is: 1/(sqrt(1-(v^2/c^2))) Where v is your velocity and c is the speed of light. So if going 99% the speed of light, each year you experience would be about 7 years to an observer on Earth (assuming I did my math right). ---
**Author:** ![Lemur866](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lemur866/32/434_2.png) [@Lemur866](https://boards.straightdope.com/u/Lemur866)\ **Post date:** [December 31, 2011, 7:44pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/4 "2011-12-31T19:44:43Z")
Except the question isn’t, “How long would it seem to take if you could go at the speed of light”, or “How long would it seem to take if you could go at 99% of the speed of light”, but rather: “How long would it seem to take (from Earth’s reference point and from the ship’s reference point) if you could accelerate at 9.8 m/s^2 halfway there, flip and decellerate at 9.8 m/s^2, flip and do the whole thing over again to get back?” To figure this out, you just need to calculate the observed times for one of the quarters of the trip, and multiply by four. Some math person will have to set up the equation for me, I used to be able to do that kind of thing… ---
**Author:** ![md2000](https://avatars.discourse-cdn.com/v4/letter/m/73ab20/32.png) [@md2000](https://boards.straightdope.com/u/md2000)\ **Post date:** [December 31, 2011, 7:49pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/5 "2011-12-31T19:49:17Z")
Basic Newton Physics: (with rounding to simple numbers) c=186,000miles/sec = 3x10^8 m/sec 1 yr = 3.2x10^7 sec 1 ly = (3.2x3) x 10^15 m = 9.6x10^15 m a=g=10 m/sec 4.3ly = (4.3x9.6)x10^15m = 4.1x10^16 m Half-way - 2.1ly = 2.0x10^16 m Time to halfway - d=(1/2)at^2 2.0x10^16=(0.5)(10)t^2 t=sqrt[(2.0/5.0) 10^16] = 6.3x10^7 sec or about 729 days. speed at turn-over v=at= (10)(6.3x10^7)=6.3x10^8m/sec Darn!! This is twice the speed of light, so I would have to look up the relativistic formula instead. If it had been at about 1/10 the speed of light or less, a Newtonian calculation would be fairly close. ---
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [December 31, 2011, 8:07pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/6 "2011-12-31T20:07:56Z")
[http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/rocket.html](http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/rocket.html) According to it, at 1g accel/decel it will take 3.6 years for you to get to the star 4.3 light years away. On Earth, about 5.2 years will pass, if I calculated it correctly. Double the numbers for getting back, obviously. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 31, 2011, 8:16pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/7 "2011-12-31T20:16:50Z")
> [@CanTak3](#): > > Assume 4.37 light years away. Obviously the fastest possible trip would be for light and would take 8.74 years round trip. But I’m more curious on two things: > > - How much time passes on earth while the human (below) makes the trip? > - How much time passes for that human since they are travelling so fast? > > So, far in the future, mankind has a spaceship with an unprecedented amount of power. The spaceship accelerates at the earths gravitational rate. It does this until it reaches the halfway mark, then the ship rotates 180º and it accelerates in the opposite direction at earths gravitational rate. When it makes a stop, it’s at Alpha Centauri (or it’s close enough!). 1. For someone on Earth’s point of view it would take the spacehip 290 million years to reach Alpha Centauri. 2. For someone on the spaceship’spoint of view it would take about 40 years to reach Alpha Centauri. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 31, 2011, 8:19pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/8 "2011-12-31T20:19:19Z")
> [@Terr](#): > > [http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/rocket.html](http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/rocket.html) > > According to it, at 1g accel/decel it will take 3.6 years for you to get to the star 4.3 light years away. On Earth, about 5.2 years will pass, if I calculated it correctly. > > Double the numbers for getting back, obviously. I have to say you’re defintely out on both those calculations. ---
**Author:** ![standingwave](https://avatars.discourse-cdn.com/v4/letter/s/9de0a6/32.png) [@standingwave](https://boards.straightdope.com/u/standingwave)\ **Post date:** [December 31, 2011, 8:25pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/9 "2011-12-31T20:25:18Z")
If you could very quickly jump to say 99.9% of the speed of light and do the same thing in reverse at the other end, it would only take a couple of months. If you could make the trip at 99.99% of light speed it would only take about 4 weeks ship time. Of course do to that requires an incredible source of energy (not to mention power) and even if you had that there would still be the crush gee forces from the acceleration so you would need one of those Star Trek inertial damper thingamajigs. ---
**Author:** ![obfusciatrist](https://avatars.discourse-cdn.com/v4/letter/o/4af34b/32.png) [@obfusciatrist](https://boards.straightdope.com/u/obfusciatrist)\ **Post date:** [December 31, 2011, 8:34pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/10 "2011-12-31T20:34:03Z")
> [@Lemur866](#): > > Except the question isn’t… Yeah, I was parsing the question incorrectly. Not sure why I wasn’t seeing what was being asked. ---
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [December 31, 2011, 8:39pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/11 "2011-12-31T20:39:17Z")
> [@These are my own pants](#): > > I have to say you’re defintely out on both those calculations. Do you think the numbers in the link are wrong? I don’t think I am out on those calcs. They look about right to me. It would take around 2 years at 1g to get to 0.9c. I think the numbers you gave in your answer are wildly wrong. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 31, 2011, 8:48pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/12 "2011-12-31T20:48:56Z")
> [@Terr](#): > > Do you think the numbers in the link are wrong? > > I don’t think I am out on those calcs. They look about right to me. It would take around 2 years at 1g to get to 0.9c. > > I think the numbers you gave in your answer are wildly wrong. The formulae in the link you’ve provided are correct, but you’ve misapplied them. I used the same formulae (thoguh I already knew what sort of order of magnitudes to expect). ---
**Author:** ![CanTak3](https://avatars.discourse-cdn.com/v4/letter/c/f14d63/32.png) [@CanTak3](https://boards.straightdope.com/u/CanTak3)\ **Post date:** [December 31, 2011, 8:52pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/13 "2011-12-31T20:52:48Z")
> [@These are my own pants](#): > > 1. For someone on Earth’s point of view it would take the spacehip 290 million years to reach Alpha Centauri. > > 2. For someone on the spaceship’spoint of view it would take about 40 years to reach Alpha Centauri. Site, formula? Using the Newtonian formula I hit the speed of light fairly quickly. I did the same math as MD2000 and realized I’d need to use relativity with it which is when I gave up. I know from one of Cecil’s own columns that for a person travelling at speed _u_ who adds speed _v_ to their total, the final output is actually _(u+v)/(1+(uv/c^2))_. But that really doesn’t help me take time dilation into account. Also, that’s a distinct change, and in reality I’m always accelerating in the OP. (Sort of the "interest compounded monthly vs. interest compounded continuously [pert]). ---
**Author:** ![DaveBfd](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DaveBfd](https://boards.straightdope.com/u/DaveBfd)\ **Post date:** [December 31, 2011, 8:54pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/14 "2011-12-31T20:54:08Z")
@Terr They’re wrong because if they were travelling at the speed of light to begin with and stopped instantly it would seem to take 4.3 years from earth. ---
**Author:** ![CanTak3](https://avatars.discourse-cdn.com/v4/letter/c/f14d63/32.png) [@CanTak3](https://boards.straightdope.com/u/CanTak3)\ **Post date:** [December 31, 2011, 9:01pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/15 "2011-12-31T21:01:52Z")
> [@These are my own pants](#): > > 1. For someone on Earth’s point of view it would take the spacehip 290 million years to reach Alpha Centauri. This can’t possibly be right. If Voyager 1 travels 1 light year every 14,000 years. The ship in the OP would be traveling much faster than Voyager 1. ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 31, 2011, 9:07pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/16 "2011-12-31T21:07:26Z")
> [@CanTak3](#): > > Site, formula? > > Using the Newtonian formula I hit the speed of light fairly quickly. I did the same math as MD2000 and realized I’d need to use relativity with it which is when I gave up. > > I know from one of Cecil’s own columns that for a person travelling at speed _u_ who adds speed _v_ to their total, the final output is actually _(u+v)/(1+(uv/c^2))_. But that really doesn’t help me take time dilation into account. Also, that’s a distinct change, and in reality I’m always accelerating in the OP. (Sort of the "interest compounded monthly vs. interest compounded continuously [pert]). I used exactly the same formula as Terr did from John Baez’s site. Like I said though I knew what to expect as I’ve doen the calculation before and alsoI believe the figures are presented (but the calculations not shown) in _A First Course in General Relatvity_ by Bernard Schutz for either this exact problem (or a very simlair one). In case someone does find an error in my calculations, I have to say I haven’t chekced them, but I’m reasonably confident they’re correct as they’re the same as what I remember they should be. ---
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [December 31, 2011, 9:11pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/17 "2011-12-31T21:11:59Z")
> [@These are my own pants](#): > > The formulae in the link you’ve provided are correct, but you’ve misapplied them. I used the same formulae (thoguh I already knew what sort of order of magnitudes to expect). The link says that the trip to 4.3 light years away, with 1 g accel/decel both ways will take 3.6 years. Did they misapply the formulas that they presented as well? ---
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [December 31, 2011, 9:13pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/18 "2011-12-31T21:13:01Z")
> [@DaveBfd](#): > > @Terr > They’re wrong because if they were travelling at the speed of light to begin with and stopped instantly it would seem to take 4.3 years from earth. Yes. And it takes longer than that at 1g. So why are they wrong? ---
**Author:** ![Asympotically\_fat](https://avatars.discourse-cdn.com/v4/letter/a/e47c2d/32.png) [@Asympotically\_fat](https://boards.straightdope.com/u/Asympotically_fat)\ **Post date:** [December 31, 2011, 9:16pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/19 "2011-12-31T21:16:10Z")
> [@CanTak3](#): > > This can’t possibly be right. If Voyager 1 travels 1 light year every 14,000 years. The ship in the OP would be traveling much faster than Voyager 1. Actually i have made an error, I will have to look in to it:o ---
**Author:** ![Terr](https://avatars.discourse-cdn.com/v4/letter/t/839c29/32.png) [@Terr](https://boards.straightdope.com/u/Terr)\ **Post date:** [December 31, 2011, 9:18pm UTC](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967/20 "2011-12-31T21:18:04Z")
> [@These are my own pants](#): > > The formulae in the link you’ve provided are correct, but you’ve misapplied them. I used the same formulae (thoguh I already knew what sort of order of magnitudes to expect). Seriously? t = sqrt[(d/c)^2 + 2d/a] c = 1 lyr/yr a = g = 1.03 lyr/yr^2 - let’s make it 1 for simplicity’s sake. d = 4.3 lyr Earth ref frame time = t = sqrt(4.3^2+2\*4.3) = sqrt(27.09) = 5.2 years. not anywhere close to 290 million years. Show me where I’m wrong. [Next page](https://boards.straightdope.com/t/anyone-remember-their-calculus-time-to-get-to-alpha-centauri/607967.md?page=2)