The balanced engine problem

[xmaslightguy]

To start, this is not based on any level of real life possibility. This is simply an exercise in "what if?" And "how so?"

So I was thinking the other day about a issue with traveling the speed of light, and that was how the mass of matter increases the closer you get to the speed of light, with it becoming infinite at c. Now I don't know the formula for this, nor how to understand it, so you'll have to find it. What I'm interested in, is would it be possible to have a perfectly designed engine that never used fuel?

Ground Rules:

- The engine can accelerate itself, the ship, and its fuel at any acceleration below c

- The engine has to use some amount of fuel

- The engine is fueled by mass, so as mass increases, so does its fuel supply

- The ship starts with 100kg of fuel, and must always have between 95 and 105 kg

- The engine can run as long as you need it to

What would acceleration of such an engine have to be?

What is its rate of fuel consumtion?

[K^2]

Relativity is relative, and that's useful to keep in mind. From perspective of people aboard the ship, the ship's mass never changes. (It's the universe around them that does weird things.) So you should immediately see why it won't work.

[Jim DiGriz]

"The engine is fueled by mass, so as mass increases, so does its fuel supply"

that's your problem right there.

there's a principle in relativistic physics that all physics is local, which means you have to look at the picture of what is going on with the object you're considering, in the reference frame of the object.

as an aside, this is why black hole singularities at the event horizon aren't really singularities. it looks like a singularity to an outside observer, but that isn't local physics. when you look at the reference frame of something falling into the black hole the singularity at the event horizon disappears and physics works fine, and you fall through the event horizon (assuming a large enough black hole you don't get ripped apart by tidal forces first). what is important here is to consider "what does the map of the universe look like to a given observer?". to an observer well outside the black hole in flat space somewhere, their 'map' has a singularity at the event horizon of a black hole. when you look at and observer falling into the black hole their 'map' does not contain the singularity at the event horizon. its the same universe, but you have to apply a change of reference frame to switch between the two views...

getting back to the mass problem...

from the perspective of an observer in the rocket, its mass does not increase. time flows normally. lengths are measured with a ruler normally. the rocket drains its fuel and pretty much works just like it does in KSP.

its the outside observer who sees the mass of the rocket increase. the problem there is that the mass of the exhaust also increases and the mass of the propellant increases, etc. you can do a translation to show that physics still works even if you do the math in the reference frame of the outside observer, but its more difficult to keep it all straight at first -- you have to increase the mass of everything in the problem, then shrink the lengths, then apply time dilation. its much easier to realize that to the kerbal in the rocket the mass will decrease exactly like it does non-relativisitically -- there is no mass increase in the reference frame travelling with the rocket. the kerbal on the rocket will step on a scale (a magical scale with a g-force of 9.8m/s that you can use in space) and measure their weight to be the same as their weight back on kerbin.

[xmaslightguy]

Wow, that explains it really well!

I recently gave an informative speech on time dilation, so I understand what you're saying.

EDIT: Hold on a sec, got a new question below...

Edited November 19, 2013 by xmaslightguy

[xmaslightguy]

Wait a second, if as the mass of the fuel increases so does the exhaust and intake, couldn't you travel faster than light if you had even the slightest amount of fuel left?

The way I see it, lets say there is a 5000 kg ship, which is accelerating at 10 m/s. Given a year, that ship will be traveling faster than light, except mass increases so it can't. At start, the F=ma formula would be 50000N=5000kg*10m/s/s (or 10m/s/s=50000N/5000kg). As mass increases, you need more force, unless what you say is true and the exhaust also increases paportionally. At some point the formula could look like 10m/s/s=(2.5*10^10N)/(2.5*10^9kg). This would keep going until velocity has reached 3*10^8 m/s, at which point the equation still works! ($ means infinity) 1m/s/s=$N/$kg, and the following second the ship would be traveling 300,000,001 m/s and thus faster than lights.

Also even if the ship used some of its mass as fuel, it would only make it accelerate faster, and so long as there is the smallest fraction of fuel left when velocity equals c, it would equal infinity and the vessel would accelerate.

What am I missing? Unless this is actually the way to travel faster than light.

[AeroEngy]

You are way over simplifying. Read this to get an idea of the equations involved. http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

Table of values of acceleration at 1g (relative to the rocket)

Note: Second column T should be a lower case t

T:time in years on the rocket.

t: time in years to a stationary observer

d: distance traveled

v: velocity at give time

Edited November 19, 2013 by AeroEngy

[Jim DiGriz]

Wait a second, if as the mass of the fuel increases so does the exhaust and intake, couldn't you travel faster than light if you had even the slightest amount of fuel left?

The way I see it, lets say there is a 5000 kg ship, which is accelerating at 10 m/s. Given a year, that ship will be traveling faster than light, except mass increases so it can't. At start, the F=ma formula would be 50000N=5000kg*10m/s/s (or 10m/s/s=50000N/5000kg). As mass increases, you need more force, unless what you say is true and the exhaust also increases paportionally. At some point the formula could look like 10m/s/s=(2.5*10^10N)/(2.5*10^9kg). This would keep going until velocity has reached 3*10^8 m/s, at which point the equation still works! ($ means infinity) 1m/s/s=$N/$kg, and the following second the ship would be traveling 300,000,001 m/s and thus faster than lights.

Also even if the ship used some of its mass as fuel, it would only make it accelerate faster, and so long as there is the smallest fraction of fuel left when velocity equals c, it would equal infinity and the vessel would accelerate.

What am I missing? Unless this is actually the way to travel faster than light.

You're also completely missing time dilation and length contraction. You're looking at the 5000kg increasing, but the 10m/s is also not constant -- the meters change and the seconds change as well.

Honestly, I did this stuff 20 years ago (I'm old) and I've forgotten how to do that. I have sat in a sophomore college physics class and showed that physics still works when you apply all three of the changes to the problem correctly, but its a bit tedious.

The rocket explanation in the sci.physics FAQ is entirely correctly but also uses hyperbolic trig functions (sinh, cosh, tanh) which makes the problem easier but is probably not very familiar. If you want to learn more about SR, by far the best intro book is:

http://www.amazon.com/Spacetime-Physics-Edwin-F-Taylor/dp/0716723271

If you read that, when you come back to that FAQ answer, it'll make more sense.