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If you have the energy for a constant 1G thrust, how long would it take to get to the planets in our solar system? How long for the 5 nearest solar systems?

Assuming turn over and decelerate at halfway.

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"1g of thrust" pointed straight up will balance gravity, and result in you floating. "1g" (as I read it), is the acceleration caused by the Earth's gravity; if that's how you actually define it, then your acceleration decreases as you get further (and 'feel less pull') from Earth. Of course, you don't need to point straight up, and TidalWave's assumption that what you meant is 9.8m/s/s is probably correct - but note that even so, his answer provides you with a minimum, eg assuming you could turn off gravity and the atmosphere (and the assumptions he mentions at the top). –  hunter2 Jul 31 '13 at 9:28
    
@hunter2, you are correct 1g of thrust will not get you off the planet. The assumption is that the starting point is in orbit, 1g of thrust during a long trip provides thrust & simulated gravity. –  James Jenkins Jul 31 '13 at 10:24
    
Fair enough. Again, his answer makes several assumptions and is a minimum (on which I'm not going to improve), but OK. // IMO, it would make more sense to use rotation for 'ship gravity' (tethered-module ship), but that's just IMO. –  hunter2 Jul 31 '13 at 10:33
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2 Answers 2

up vote 5 down vote accepted

Assuming acceleration is constant, $d=(1/2) a t^2$. So plotted over time, distance traveled is a nice parabola.

If you want the time it'd take for a specific distance, it's easy to manipulate $d=(1/2) a t^2$.

$t=\sqrt{2d/a}$

If you're using meters and seconds as your units, $a=9.8 meters/sec^2$

To travel half the distance to the moon would take about 1.75 hours. The other half distance spent decelerating would take the same amount of time.

enter image description here

Using Days and AU (astronomical units) we can see 3 days will get about 2.5 AU (halfway to Jupiter). 4.5 days will get you 5 AU (halfway to Saturn). 9 days will get you 20 AU (more than halfway to the Kuiper belt)

enter image description here

It gets trickier for interstellar distances. In Newtonian mechanics v = at, so it'd take a little less than a year to reach c at 1 g acceleration. But relativity won't allow that, we can only get close to c.

Our Newtonian model is okay for nearly a year of acceleration and after that relativity wrecks this nice parabola:

enter image description here

After 1 year at 1 g we will have traveled .5 lightyears and our velocity will be close to maxed out. There after we're moving at close to c, so add a little more than a year for each lightyear distance.

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Your "add a little more than a year for each lightyear distance" is correct for an outside observer, but for someone aboard the ship, the Newtonian model is correct for all distances (as measured before starting acceleration): Lorentz contraction will shrink the universe during travel to give the appearance of Newtonian physics. –  Mark 21 hours ago
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According to wikipedia, interstellar travel at 1G would take approximately 1 year + the distance in lightyears. Proxima Centauri (4.2 light years) for example would take 5.2 years.

But that time is from the viewpoint of stationary observers at the departure point. The trip's duration from the traveler's viewpoint would be less due to the time dilation effect predicted by Einstein's Theory of Relativity. The greater the distance, the greater the speed from the stationary observer's viewpoint. From the stationary observer's viewpoint the traveler's rate of acceleration would slow as they approached the speed of light. The traveler would see no change between their speed and the speed of light. Instead they would experience time at an increasingly slower rate which would effectively cause the distance to the destination to become shorter.

Due to the time dilation effect, 1G acceleration should be sufficient to travel anywhere in our galaxy in less than a lifetime from the viewpoint of the traveler, but not the stationary observer.

For more information on the time dilation effect read Stephen Hawking's "Brief History of Time"

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Wikipedia article on constant acceleration interstellar travel: en.wikipedia.org/wiki/Space_travel_using_constant_acceleration –  Anthony X Jul 20 at 17:14
    
It should probably be mentioned that it becomes increasingly more difficult to sustain constant acceleration as your speed approaches a significant portion of c. I did briefly mention relativistic effects in my answer, but there are literally infinite number of problems to consider at speeds close to c for anything with a rest mass larger than zero, including e.g. non-uniform distribution of matter between points A and B that additionally causes variations in time dilation, and interaction with any matter at relative velocity at or over c causes a series of other relativistic effects. –  TildalWave Jul 20 at 17:31
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@TildalWave what happened to your answer? It was a very good answer with a lot of information. –  HopDavid yesterday
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