I Farthest distance a human can travel from Earth in a life

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1. Feb 28, 2017

James Horner

I've been stuck on this physics problem for several years now. I would be very grateful if someone could explain how to solve this problem. The farthest a human could travel from Earth in one lifetime theoretically is limited only by the acceleration a human can withstand, and the length of their life. Solving the problem below would answer this question, taking into account relativity.

A spaceship starts at rest on Earth, ignoring gravity. The spaceship then accelerates, and from the perspective of the astronauts inside the spaceship, it appears to accelerate at a constant 9.8 m/s^2. After the astronauts on the spaceship get 100 years older, how far away will the spaceship be from the perspective of Earth?

2. Feb 28, 2017

Comeback City

If you want to keep this simple and ignore the maximum physically possible velocity of the spaceship, then this can be easily solved using a displacement equation:
Δx=v0Δt + (1/2)aΔt2
where
Δx is displacement
v0 is initial speed
a is acceleration
Δt is time elapsed

3. Feb 28, 2017

James Horner

I understand classical acceleration equations, my question is how far the spaceship would be taking into account time dilation, length contraction, ext...

4. Feb 28, 2017

Staff: Mentor

In other words, ignore the fact that the ship can't go faster than light? What's the point of ignoring the actual laws of physics that apply to the problem?

The best quick resource is the Usenet Physics FAQ article on the relativistic rocket equation:

http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

It has everything you need.

5. Feb 28, 2017

pervect

Staff Emeritus
Try googling for "The Relativistic Rocket", for example <<this link>>.

In 28 years, if you don't want to stop, you can reach Andromeda, accelerating at 1g.

In 100 years, if you don't slow down to stop at the destination, using the special relativistic (SR) formula, I make the distance you cover to be on the order of 10^22 light years. But you'd undoubtedly need to take into account the expansion of the universe as the FAQ mentions, and I don't see any discussion of the needed formula to do this. And it would be tricky to work it out correctly, amongst other issues one would need to be clear on how the distance was being measured in the first place.

Using the same SR formula, if you wanted to stop at the destination, the distance you'd cover in 100 years would be on the order of 3*10^11 light years or so. Again, this doesn't have the needed GR corrections.

(add) As peter mentioned (he types faster), the SR number DOES take into account the speed-of-light limit, and other relativistic effects such as time dilation. As you'll probably notice, the distance you can cover according to SR is considerably larger than the distance you could cover in a Newtonian model.

6. Feb 28, 2017

Comeback City

A simple mental error led me to believe it couldn't reach light speed in 100 years. Indeed, I was wrong, as the speed reached would be about 3.09E10 m/s.

7. Feb 28, 2017

Staff: Mentor

As a rule of thumb, a 1 g acceleration means getting close to light speed in about 1 year. The time varies inversely with the acceleration, so a 100 g acceleration would mean getting close to light speed in about 1/100 of a year (or a few days), and a 0.01 g acceleration (within range of today's ion drives) would mean getting close to light speed in about 100 years.

8. Feb 28, 2017

Comeback City

The article is very interesting!

9. Feb 28, 2017

PAllen

That is a useless answer because this problem is extremely relativistic. The answer is about 2.6 * 1044 light years. See:
http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

10. Feb 28, 2017

Comeback City

PeterDonis got it covered.

11. Feb 28, 2017

PAllen

I get a different answer than Pervect. I've double checked mine and it seems right.

12. Feb 28, 2017

James Horner

Thanks for the help, the relativistic rocket is a really helpful article. I'm still trying to figure out which exact equation describes the distance as a function of acceleration and time.

13. Feb 28, 2017

Staff: Mentor

It's the equation for $d$ (lower case) in terms of $a$ (the acceleration) and $T$ (upper case--that's the time according to the crew of the rocket).

14. Feb 28, 2017

DrGreg

I agree with your answer$$\frac{\cosh(103) - 1}{1.03} = 2.62 \times 10^{44} \text{ light years}$$

15. Feb 28, 2017

Comeback City

There is a slight difference between your answers. Did you both solve it a different way or something of the sort?

16. Feb 28, 2017

DrGreg

Sorry, I made a typo which I have now corrected. 44 not 43.

17. Feb 28, 2017

Comeback City

Oh okay I see now.

18. Feb 28, 2017

rootone

That sounds almost like something doable with present technology and some gaffer tape!

19. Feb 28, 2017

Comeback City

Quick question: what equation is this that you used (I understand the hyperbolic cosine part just not the whole thing)? Is it related to the Tsiolkovsky rocket equation?

20. Feb 28, 2017

PAllen

Peter already answered this in post #13. Everything you need to check this is that equation and constants in useful units provided in that rocket article. If you do the google search Pervect suggested, you will also find several links to derivations of the relativistic rocket equations from first principles.