Calculating speed / distance over time?

[adt2]

Hiya, folks. I've got what may or may not be a high-school level physics question, but frankly I've never taken a physics class and even more frankly, I'm not sure what terms to search for on Google to find the answer myself.

Consider the following hypothetical circumstances. A spaceship, starting from low-earth orbit, accelerates toward a distant planet (or star or whatever). Assuming the craft accelerates at a constant rate indefinitely (I know, I know - just humor me), how long will it take to get where it's going?

If the craft was headed somewhere, say, 15 light-years away, how long would it take to get there? I assume it would have to decelerate for just as long as it accelerated in order to come to a stop at its destination, so there should be 50% acceleration and 50% deceleration (negative acceleration?) in total, right?

This concept resulted from a conversation with my 10-year-old son, and I told him I'd figure it out, but clearly I'm not as smart as I thought I was. Any help will be appreciated!

 

[maajdl]

It depends a lot on the type of fuel used.
As the spaceship accelerate it will lose the weight of the material leaving the reactor.
Remember the size of the complete Apollo rocket just to bring a relatively small lunar module on the moon.
If -by some miraculous technology- the weight loss would be negligible, they the calculation is easier.
If the acceleration is large, then it is even much easier: we can neglect the time for the spaceship to reach close to the speed of light and therefore it will be of the order of 15 years. It can take any longer as one want if we like the landscape.
These 15 years is the time we would infer by communicating we him and using our own clock, and it will take us 30 years or more before we now he reached the destination.

However, if the spaceship really accelerated close to the speed of light, then the journey as measured by clocks on board could have been much faster: a month, a week, a day, or even a second if the acceleration did not destroy the spaceship. That's what we learned 109 year ago from Einstein and that has been verified experimentally.

 

[adt2]

So let's make some assumptions:

• The ship neither gains nor loses weight (mass?) during the trip. Say it uses solar power or space dust or radiation waves or something to create thrust.
• The rate of acceleration is reasonable - something we might be able to actually accomplish with existing or near-future technology (i.e. not warp or wormholes or light-speed). Voyager II launched in 1977 and is currently traveling something like 39,600 mph. If we disregard the acceleration required to leave earth, and instead start from low-earth orbit, how fast would Voyager II be traveling right now if its rate of acceleration had remained constant since leaving Earth's atmosphere?

Part of my problem is that I have no idea what's "reasonable" with respect to rate of acceleration away from low-earth orbit. If the Space Shuttle Orbiter pointed its nose toward Jupiter and fired whatever thrusters it has at full burn, how fast would it accelerate?

 

[boneh3ad]

Basically you are asking if the spacecraft started from Earth and accelerated in a straight line uniformly toward an object, how long would it take?

If that's the case, you can use the basic kinematic equations for constant acceleration, often called the SUVAT equations. You can derive those pretty simply using basic calculus, too, if you know any of that.

Basically the equation in question is number 2 in that link,
$$r = r_0 + v_0 t + \dfrac{1}{2}a t^2.$$

From that, assuming you start with zero initial velocity and count the starting position as zero, then the time it would take to get to some point a distance $r$ away from the Earth would be
$$t = \sqrt{\dfrac{2r}{a}}.$$

That's about the most simple estimate you can get for what I think you are asking. You can add essentially infinite complexity to that if you want, obviously.

 

[jtbell]

You're going to have to take relativity into account. For example, see this page:

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

It may have the equation for calculating what you're looking for. A Google search on "relativistic rocket" turns up other pages on the same topic.

 

[adt2]

Holy mackerel. I was hoping for something like "a squared + b squared = c squared," or maybe a quick Excel formula. I'm starting to remember why I never took any physics classes.