Farthest distance a human can travel from Earth in a life

[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?

 

[Comeback City]

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?

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

 

[James Horner]

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

 

[PeterDonis]

If you want to keep this simple and ignore the maximum physically possible velocity of the spaceship

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?

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

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.

 

[pervect]

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?

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.

 

[Comeback City]

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?

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.

 

[PeterDonis]

A simple mental error led me to believe it couldn't reach light speed in 100 years.

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.

 

[Comeback City]

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.

The article is very interesting!

 

[PAllen]

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

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

 

[Comeback City]

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

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.

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.

PeterDonis got it covered.

 

[PAllen]

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

 

[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.

 

[PeterDonis]

I'm still trying to figure out which exact equation describes the distance as a function of acceleration and time.

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).

 

[DrGreg]

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

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

 

[Comeback City]

That is a useless answer because this problem is extremely relativistic. The answer is about 2.6 * 10^44 light years. See:
http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

I agree with your answer

cosh(103)−11.03=2.62×10^43 light years​

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

 

[DrGreg]

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

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

 

[Comeback City]

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

Oh okay I see now.

 

[rootone]

As a rule of thumb, a 1 g acceleration means getting close to light speed in about 1 year.

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

 

[Comeback City]

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

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?

 

[PAllen]

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?

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.

 

[pervect]

I'm not sure where I went wrong, but if Peter & Dr. Greg both get different answers, I probably made an error somewhere.

 

[PeterDonis]

if Peter & Dr. Greg both get different answers

It was actually PAllen who posted the same answer as DrGreg, but I'll quickly run through a sanity check. An acceleration of 1 g is nice because it equates to almost 1 in units of years and light years (as the Usenet Physics FAQ article notes, 1 g = 1.03 lyr/yr^2). So we can rewrite the equation for $d$ in a much simpler form:

$$
d = \cosh T - 1
$$

since both $c^2 / a$ and $a / c$ are approximately 1. So we just need to evaluate $\cosh 100 - 1$. The $1$ will be negligible, and my calculator says $\cosh 100 = 1.344 \times 10^{43}$. This will be the distance in light-years that can be reached in 100 years of ship time.

That is for a journey where we don't want to stop at the destination. If we do, we need to cut $T$ in half, since we will have to spend half our time accelerating and half our time decelerating; then we multiply $d$ by two for the two legs of the journey. So we will have

$$
d = 2 \left( \cosh \frac{T}{2} - 1 \right)
$$

This gives (the 1 is still negligible) $2 \cosh 50$, which my calculator says is $5.185 \times 10^{21}$ light-years.

DrGreg and PAllen got slightly larger answers because they used 1.03 for the acceleration instead of the 1 I used here as an approximation, which increases the argument of $\cosh$ enough to increase the answer by about an order of magnitude. So their answers look correct to me.