Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: thedoc on 01/05/2013 10:30:01
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Jack Stott asked the Naked Scientists:
Dear Dr Chris.
I have plucked up the courage to write to you with a question from a colleague of mine which I can't answer.
I have no idea why he wants to know this ( I think he reads too many science fiction comics or watches too much 'Star Treck' ) but here goes :-
If an object of negligible size & mass is launched from a standing start in a vacuum, and is subjected to an acceleration force of 1 g - how long will it take to reach the speed of light.
Hope you are able to provide an answer, or even a formula to calculate an approximate result when and if you have the time.
Best Regards
Jack Stott BSc(Hon) Elec Eng Science
What do you think?
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It will never reach the speed of light. It would take an infinite amount of energy. This results from Einstein's theory of Special Relativity. However, from the perspective of anyone travelling on this craft, they will experience a contraction in the distances in their direction of travel. So although they can never get to 186,000 miles per second they can (theoretically), nonetheless, achieve a speed such that they can travel a distance that they may initially have measured (before accelerating) as 186,000 miles in less that 1 second as measured on their clocks. If I remember correctly this turns out to be the same time (to get to this speed) as would be calculated by Newtonian mechanics - I would need to check this with some maths to be sure. There are some unfortunate consequences of travelling this fast resulting from time dilation so that should return at some point you would find the earth you left having aged considerably compared to yourself.
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There are many unfortunate consequences of travelling near the speed of light, microscopic specks of dust would be like express trains when you hit them and the CMBR would be blue shifted up to Gamma rays.
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Yes. Forward Shields to maximum please Mr Scott :-)
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Jack Stott asked the Naked Scientists:
Dear Dr Chris.
I have plucked up the courage to write to you with a question from a colleague of mine which I can't answer.
I have no idea why he wants to know this ( I think he reads too many science fiction comics or watches too much 'Star Treck' ) but here goes :-
If an object of negligible size & mass is launched from a standing start in a vacuum, and is subjected to an acceleration force of 1 G - how long will it take to reach the speed of light.
Hope you are able to provide an answer, or even a formula to calculate an approximate result when and if you have the time.
Best Regards
Jack Stott BSc(Hon) Elec Eng Science
What do you think?
Wonderful question, the answer depends on the details, if "negligible" mass means zero then the particle is already moving at c and can not be at a standing start in any (inertial) reference frame. If not, then the velocity can be computed from the equations of Einstein's special theory of relativity, rather than repeat, please refer to this link for the details: http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html (http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html)
Since we are assuming an acceleration of 1g, the size and mass does not enter into the velocity calculation, it will matter in terms of the energy required to accelerate the particle. So, after 1 year at 1g, 0.77 of the speed of light, 2 years, 0.97c, 12 years to get to 0.99999999996, pretty close to c but not close enough for a physicist :-)
Hope this helps, a simplified answer to a sophisticated question. Science fiction sometimes becomes science fact, keep the wonder!
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Given a proper acceleration "a" (the acceleration felt by the astronaut inside his ship) and denoted with "T" the proper time (the time measured by the astronaut's wristwatch), the spaceship' speed u, as measured from Earth, is given by:
u(T) = c [exp(aT/c) - exp(-aT/c)] / [exp(aT/c) + exp(-aT/c)]
Example 1:
a = 1g ~= 10m/s2
T = 347.22 days ~= 30,000,000 s
c ~= 300,000,000 m/s
--> u = c [e - 1/e] / [e + 1/e] ~= 0.76 c
Example 2:
a = 1g ~= 10m/s2
T = 9.5 years ~= 300,000,000 s
c ~= 300,000,000 m/s
--> u = c [e10 - (1/e)10] / [e10 + (1/e)10] ~= 0,999999996c.
As others have pointed out, u becomes = c only when T = +oo.
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As stated above a particle subjected to uniform accleration as measured in its own frame of reference wil never travel at the speed of light. It will only come closer and closer to it. On the other hand its impossible to force a particle to accelerate at a constant coordinate acceleration, i.e. as measured in a particular inertial frame of reference. E.g. place a charged particle in a uniform electric field then the force on the charge would be constant but its acceleration would decrease with time. You can think of this as the (inertial/relativistic) mass increasing with speed and thus with time.
I worked out this derivation here
http://home.comcast.net/~peter.m.brown/sr/uniform_accel.htm
There is a nice spacetime diagram which illustrates the particle's position as a function of time. It should make all this clearer.
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Very nice Pete.
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All that relativity garbage is just that. Garbage. As you aproach the speed of light nothing changes. It doesn't take any longer to accelerate the last half of the way to the speed of light as it does the first half. When you acheive the speed of light nothing happens , nothing changes. If you keep accelerating at 1G twice as long as that then you will be going twice the speed of light. Specks of dust should be destroying the space station by now because for all we know the solar system and the entire galaxy could be moving away or toward one another at 100s of times the speed of light. Unless your Christopher Colombuses Queen of Spain and you beleive the ocean continues on to infinity and where your sitting is the center of the universe, the sun goes around the earth and this is ground zero and you are perfectly still. Did you measure our speed in the universe by the warehouse walls at the edge morons? What I want to know is where these idioitic concepts like Vaporizing or going back in time come from? Since you haven't invented a spacecraft that can accelerate constantly at a 1G acceleration you automatically asume from some erouneous candle light and horse era mathimatical scribble that you will vaporize at the speed of light which if you recall they also tried to sell us about breaking the SOUND barrier before it was possible. Why do you mathematical types persist at this nonsense. It takes 380 something days to accelerate to and PAST the speed of light as closely as I had calculated at one time but I cannot ever get a straight answer on this very pertinent question because of all the geeknoid scientific daydreaming about these infinite forces barring coming near the speed of light GETTING IN THE WAY. Shut up all ready you have no proof for all that hogwash.
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Not worded quite correctly the post meant to compare the solar system and this galaxy moving toward or away from other galaxies (not within our own)at possibly 100s of times the speed of light.
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All that relativity garbage is just that. Garbage. As you aproach the speed of light nothing changes.
Nope , time dilates ... http://en.wikipedia.org/wiki/Time_dilation#Time_dilation_due_to_relative_velocity
There is experimental evidence for it, e.g ... "Experimental Evidence for Time Dilation: Dying Muons (http://galileo.phys.virginia.edu/classes/252/srelwhat.html)"
http://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System#Relativity
http://en.wikipedia.org/wiki/Time_dilation#Experimental_confirmation
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There is some extra weirdness that happens close to the speed of light. I read an article recently about two ships traveling close to the speed of light attached by a rope....does it break or no. Length contraction and other effects.
I didn't quite understand it, it was a bit involved...just thought id add that in.
Bell's spaceship paradox feel free to knock yourself out on this one, lol
http://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox
From the speedy travelers perspective there appears to be no upper bound or speed limit. You keep on going faster and faster. Time dilation effects, you can travel anywhere in the observable universe in a short space of time, by your watch anyway.
I think a number of unpleasant side effects start to accumulate. This might mean that traveling very close to the speed of light might be impractical even if energy for propulsion wasn't an issue.
Shielding solution.
Imagine a giant cargo net full of ice floating just ahead of your speeding ship. This should take care of small dust grains and EM radiation. Space is cold so machine gunning that snowball and lobbing in the odd artillery shell isn't to much of a worry so long as the ice pack can dissipate the heat fast enough. Quite a few layers of steeply angled metal Armour also help.
A very powerful forward facing laser might also help. Heating a corridor of space directly in your path. This will tend to make the dust particles scatter out of the way. Assuming no weird ionization effects etc cause the opposite to happen.
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There is some weirdness that happens close to the speed of light. I read an article recently about two ships traveling close to the speed of light attached by a rope....does it break or no. Length contraction and other effects.
As you said, it's called Bell's Spaceship Paradox. I know the problem well. See
http://en.wikipedia.org/wiki/Bell's_spaceship_paradox
http://math.ucr.edu/home/baez/physics/Relativity/SR/spaceship_puzzle.html
I didn't quite understand it, it was a bit involved...just thought id add that in.
It's easy to understand once it's properly explained. Consider two spaceships which start out from rest in the inertial frame S. Connect them with a piece of string which is strung taught. Let each of them accelerate at the exact same rate so that as measured in S they maintain the exact same distance apart. Since the string also undergoes acceleration its speed is constantly increasing. Therefore its length must undergo a Lorentz contraction. Eventually the spring will break. Consider this from the Spaceship's point of view. Each astronaut sees the other one initially at rest with respect to each other. Their accelerations are the same so they must be at the same distance apart, right? Therefore the string doesn’t break. That's the paradox.
The resolution to the paradox is to consider the spaceship which is trailing behind to be at rest in a uniform gravitational field and the other ship being above him in the gravitational field with his rocket's turned on with the exact same amount of thrust. Since the local acceleration in a uniform gravitational field decreases as one goes higher in the gravitational field the astronaut in the higher position, who has his rocket engines doing the same thing as the one lower in the field, must be accelerating away from the spaceship below him. Since he's moving away the string connecting them breaks. That’s the resolution to the paradox.
Make sense?
From the speedy travelers perspective there appears to be no upper bound or speed limit. You keep on going faster and faster. Time dilation effects, you can travel anywhere in the observable universe in a short space of time, by your watch anyway.
If the traveler feels the same acceleration in his frame of reference then his acceleration as measured from an inertial frame of reference will decrease with speed at such a rate that his speed approaches the speed of light but never gets there.
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It's easy to understand once it's properly explained. Consider two spaceships which start out from rest in the inertial frame S. Connect them with a piece of string which is strung taught. Let each of them accelerate at the exact same rate so that as measured in S they maintain the exact same distance apart. Since the string also undergoes acceleration its speed is constantly increasing. Therefore its length must undergo a Lorentz contraction.
Lorentz contraction in which frame of reference? Not in S, since "as measured in S they maintain the exact same distance apart".
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All that relativity garbage is just that. Garbage.
Wrong. You’re basing that on the way you imagine that it should be which is grounded in every day experience. But everyday experience does not give you any sense of what happens at high speed. If you were to base your assumptions on what really happens rather than what you think happens you’d be singing another tune.
As you aproach the speed of light nothing changes. It doesn't take any longer to accelerate the last half of the way to the speed of light as it does the first half. When you acheive the speed of light nothing happens , nothing changes. If you keep accelerating at 1G twice as long as that then you will be going twice the speed of light.
That’s been proven to be wrong. We know how things work under constant proper acceleration because it happens in particle accelerator labs all the time. Theory and experiment correspond to exactly what is predicted.
Since you haven't invented a spacecraft that can accelerate constantly at a 1G acceleration you automatically assume…
Wrong. We can accelerate things at constant proper acceleration. It’s done with particles all the time. E.g. if you place a charged particle in an uniform electric field then the force on it is constant and in a frame of reference which is momentarily at rest with respect to it the accelerate has a constant value. In the lab frame the acceleration decreases with time and the particle gets closer and closer to the speed of light in a manner predicted exactly (to experimental accuracy) by the theory of relativity
Why do you mathematical types persist at this nonsense.
We’re not mathematical types. We’re physicist types who use math to describe the physics. And because that’s what’s observed in the laboratory. Why do you non-mathematical types refuse to learn what actually happens rather than what you think happens? And why do people like you refuse to learn the physics?
Shut up all ready you have no proof for all that hogwash.
Why do you think being rude will help you convince people who know what actually happens that it doesn’t actually happen?
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Lorentz contraction in which frame of reference? Not in S, since "as measured in S they maintain the exact same distance apart".
Lorentz contracted in the frame in which the string is moving. When the string tries to Lorentz contract it breals because in its own istantaneous frame the spaceships are moving apart. I explained all that above.
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It does, as long as we give a 'inertial' preference to the ship under Pete. But what happens if we define the ship above as being 'inertial' too, at rest in a gravitational field? And if we assume the ship above to be at rest, with the ship under going for it at , ahem, full throttle (one constant uniform G acceleration) it becomes really intriguing :)
Then again, what about the contraction you observe, relative a distance measured in front of you. The ship 'under' should also see the space contracted in front of it, including the space between the ships as I think? What about the ship above, looking back, at the ship 'below', would they agree to it being a same distance between them?
As they are defined as being 'at rest' with each other? What would 'at rest' mean there?
Different constant distances?
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Assuming a length contraction to be a symmetry in both directions (backward as well as forward of the ships motion) you can define them as being at rest, and that should make the best sense, possibly :) but you have a acceleration of matter to consider too, as well as the rope. A little like the (not infinitely) rigid pole you move, poking at the moon, with the 'motion propagating' in the rod at a approximate speed of sound, compressing and decompressing itself. Here you have particles of that matter constantly compressing (and decompressing?) relative each other as the 'force' of one G 'propagates in the matter, or should I assume that they are in a constant uniform state of compression? What I mean is that I find it tough to see that matter as being in a equilibrium as it is accelerated.
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It does, as long as we give a 'inertial' preference to the ship under Pete. But what happens if we define the ship above as being 'inertial' too, at rest in a gravitational field? And if we assume the ship above to be at rest, with the ship under going for it at , ahem, full throttle (one constant uniform G acceleration) it becomes really intriguing :)
Then again, what about the contraction you observe, relative a distance measured in front of you. The ship 'under' should also see the space contracted in front of it, including the space between the ships as I think? What about the ship above, looking back, at the ship 'below', would they agree to it being a same distance between them?
As they are defined as being 'at rest' with each other? What would 'at rest' mean there?
Different constant distances?
There is a difference between the ships. One ship is lower in the gravitational field that the accelerating frame mimics while the other is higher up. This is what breaks the symmetry. The clock in the higher up (i.e. leading ship) has its clock running faster than the one lower in the field (i.e. the trailing ship).
Did you understand everything I was trying to explain when it came to looking at this all from the perspective of a gravitational field?
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Think so :)
You're defining the ship 'under' as being 'at rest', presenting a uniform gravitational field, aka a 'inertial object' (loosely expressed a planet of some mass). You then go to define the other ship as constantly accelerating in that gravitational field. It being at a higher altitude versus that 'field', so giving it a 'faster clock' relative the 'planetary gravity' represented by the ship under. The problem is, to me, that it seems that I can use any of the ships for such a definition, or assume both to be in a uniform gravitational field there?
And doing so, instead defining the one 'above' as presenting this uniform gravity, you get a result in where the ship 'under' is uniformly constantly accelerating into that field? Although it still is a symmetric solution, as you have one ship at a different 'height' relative that field, no matter how you define them, it becomes slightly confusing to me.
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I don't think one can define the ship 'above' as 'hovering' at a constant height, using the first example of yours? As that wouldn't give me a equivalence with the situation of two ships constantly uniformly accelerating, as you have a ever growing expenditure of energy as the ships continue to accelerate?
I'm not sure there?
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Think so :)
You're defining the ship 'under' as being 'at rest', presenting a uniform gravitational field, aka a 'inertial object' (loosely expressed a planet of some mass). You then go to define the other ship as constantly accelerating in that gravitational field.
No. My neck hurts from typing or so long so I'll explain later.
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Ok Pete, I'll read it with interest.
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Is this solution using two exactly equivalent ships Pete? Everything the exact same for both ships? As mass, acceleration, etc? If I assume them them being 'exact replicas' of each other in every aspect, accelerating equivalently, in a equivalently 'flat space', then they also should be able to be described as belonging to a same frame of reference I think? Wouldn't such a definition make their 'clock-ticks' equivalent too? And if they are, how do I from that get to different heights?
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Lorentz contraction in which frame of reference? Not in S, since "as measured in S they maintain the exact same distance apart".
Lorentz contracted in the frame in which the string is moving. When the string tries to Lorentz contract it breals because in its own istantaneous frame the spaceships are moving apart. I explained all that above.
Quoting from the wiki page that you have linked:
<<In the inertial frame S, a delicate string or thread hangs between two identically accelerating spaceships.>>
Where does it say that "as measured in S they maintain the exact same distance apart"? If they maintained the same distance apart, the string wouldn't broke.
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Where does it say that "as measured in S they maintain the exact same distance apart"?
The fact that the ships accelerate at the same rate and thus remain the same distance apart is the entire point of the paradox. In fact that’s what the paradox is all about.
Please read it again because it states
The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start, because in S it is effectively defined to remain the same, due to the same acceleration of both spaceships.
I don’t understand what the problem your having is. It’s very simple. As measured in an inertial frame S (in flat spacetime) if two ships accelerate at the exact same rate then they must remain the exact same distance apart at all times as measured in frame S. If you don’t know that they you don’t know the problem at hand because this is what the problem is all about.
If they maintained the same distance apart, the string wouldn't broke.
That's quite wrong. Consider a string whose proper is L0. The proper length of an object is defined to be the length of the object as measured in the frame in which the object is at rest.
In the inertial frame S string is moving parallel to its length. In S the string has undergone a Lorentz contraction and as such it will be shortened to the length L = L0*sqrt(1 - v2/c2). Since the ships accelerate at exactly the same rate the remain the same distance apart in the frame S. Therefore after the acceleration has ended and they’re moving at constant speed the distance between them as measured in S will be greater than they were before they started moving. This means that they will be further apart than when they started. Since the properties of the string never changed the string had to break because the ends were moved apart.
QED
Any questions now?
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Where does it say that "as measured in S they maintain the exact same distance apart"?
The fact that the ships accelerate at the same rate and thus remain the same distance apart is the entire point of the paradox. In fact that’s what the paradox is all about.
Please read it again because it states
The distance between the spaceships does not undergo Lorentz contraction with respect to the distance at the start, because in S it is effectively defined to remain the same, due to the same acceleration of both spaceships.
I don’t understand what the problem your having is. It’s very simple. As measured in an inertial frame S (in flat spacetime) if two ships accelerate at the exact same rate then they must remain the exact same distance apart at all times as measured in frame S. If you don’t know that they you don’t know the problem at hand because this is what the problem is all about.
If they maintained the same distance apart, the string wouldn't broke.
That's quite wrong. Consider a string whose proper is L0. The proper length of an object is defined to be the length of the object as measured in the frame in which the object is at rest.
In the inertial frame S string is moving parallel to its length. In S the string has undergone a Lorentz contraction and as such it will be shortened to the length L = L0*sqrt(1 - v2/c2). Since the ships accelerate at exactly the same rate the remain the same distance apart in the frame S. Therefore after the acceleration has ended and they’re moving at constant speed the distance between them as measured in S will be greater than they were before they started moving. This means that they will be further apart than when they started. Since the properties of the string never changed the string had to break because the ends were moved apart.
QED
Any questions now?
Ok, I think to have understood it now, thank you very much!
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lightarrow
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Any questions now?
Ok, I think to have understood it now, thank you very much!
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lightarrow
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I don't understand something. The way this article appears listed it seems as if lightarrow posted last in the thread. When I looked at it when I read the thread it no longer appears that way. What's going on here?
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I don't understand something. The way this article appears listed it seems as if lightarrow posted last in the thread. When I looked at it when I read the thread it no longer appears that way. What's going on here?
It's a relativistic thread. Observers don't necessarily agree on which posting event comes first.
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I don't understand something. The way this article appears listed it seems as if lightarrow posted last in the thread. When I looked at it when I read the thread it no longer appears that way. What's going on here?
It's a relativistic thread. Observers don't necessarily agree on which posting event comes first.
Lol!! Okay. I have to admit. That's a good-un! :D
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I'm not sure Pete, to me it boils down to if a Lorentz contraction is a complementary effect to a time dilation, depending on observer. If it's not complementary then you make eminent sense. If it is though, you still have to explain how a same frame of reference can give us different 'clock ticks'? Because that is how I read such a description, everything being 'equivalent', a 'flat space', 'exact replicas' of ships, etc. I would call it a same frame of reference, relative the ships involved?
Maybe you could argue, as it is a acceleration involved with one ship 'leading', the other 'trailing', that this 'same frame of reference' I'm alluding too isn't correct? So let me ask a question, can one expect two, exactly equivalent frames of reference, to share a same ruler and clock?
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Like this possibly? One might assume the leading ship at each point of 'space' to present the exact same 'increasing acceleration' as the other 'trailing' ship have, in each moment, although later crossing that exact point having a further thrust than the first. (Considering as 'energy expended' now as they constantly accelerate.) Making the definition of one singular frame of reference for both wrong. And then we have 'simultaneity' too?
Even if we assume both ships to have the exact same acceleration, using them crossing some fixed point in space relative that acceleration, the trailing ship must then give us another answer in form of energy expended, from the first, if you see how I'm thinking :) Tricky this one.
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Nah. Can't make that one work for me.
Think the point is that if they are indeed sharing a same frame of reference (rest frame) they you should be getting the same answers, relative their local clock and ruler. So if they do not, then neither will they share a same frame of reference. And using the last definition, then the string must break.
The first description demands us to ignore whatever distance traveled, relative passing some SpaceTime position for each, instead looking at what is equivalent at each moment in 'SpaceTime'. If everything would be equivalent then no string should break, as I think.
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So what it boils down too for me, is indeed, can I assume equivalent frames of reference to also share a same clock and ruler? If it can, then either this thought experiment fails to me as a idea, as it includes acceleration instead of uniform motion. Or, the string won't break as we now have moved from a acceleration, through (passing) SpaceTime points, instead using the 'eyes of a God', able to define two equivalently accelerating objects as being 'still', belonging to one same 'frame of reference'.
anyway I look at it this one hurts my head :)
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I don't understand something. The way this article appears listed it seems as if lightarrow posted last in the thread. When I looked at it when I read the thread it no longer appears that way. What's going on here?
It's the same question I asked myself. I wrote you twice an answer to your post, in which I was thanking you for your explanation of the problem, but it didn't appear. Let's see this time...
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lightarrow
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under 4 days to reach 10% the speed of light. at 10meters per second^2 acceleration.
here's my off the thumb calculation . light is 300 million meters per second speed. 10% of this is 3 million meters per second. gravity at 10 meters secone ^2 will get you to 3 million in 300,000 seconds.
1 hour is 3600 seconds. 300 000 / 3600 is 300/3.6 ----83.3 hours. which is 3 dyas 11.3 hours. again---i used 10 instead of 9.8 for the speed of acceleration to make things simpler. to because the real acceleration is 2% slower.
it's somewhere close.
i would say under 4 days. if you wanna do the full calculation go for it.
10% the speed of light is fast as sh1t and very few celestial objects, as opposed to plasma winds, have been observed to move this fast. 3million meters per second is 3000 kilomters per second. the fastest asteroids appraoching earth, the few outliers beyond the 3 standard devitations are between 50 and 60 km/s . the vast majority 98% are between 30km/s and over 4km/s.
3000km/s is about 100 times faster the routinely fastest asteroids passing by earth.
mercury, the fastest moving planet by far is 48km/s relative to revolving around the sun.
mercuries orbit is influenced by relativistic effects due to the suns strong graviational field. and it's speed is only 48km/sec. the earth moves in the low 30's relative to the sun.
the fastest object ever launched from earth was about 18 km/sec relative to earth and thus almost as fast as mercury relative to the sun. we are a LONG LONG way off getting to 3000/km/sec relative to the earth, or the sun.
and remember, anytime you are launching in our solar system away from the sun. you are fighting the sun's gravity, which varies with distance of course. .
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Ig = 9.8 m/sec/sec = 0.588 km/min/min = 35.28 km/hr/hr = 2116.8 km/day/day
the speed of light is around 300,000 km/sec, so after 141 days you would be approaching the speed of light if you were constantly accelerating at 1 g.
It's no big deal accelerating at 1 g, we are doing that all the time just lying in bed, but it would be a big deal if we accelerated away from the Earth at 1 g constantly because we would have to find a source of energy to do so.
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Over 3 years and dozens of replies and then finally someone who doesn't have their head stuck up their ..... finally answers the question. LOL.
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Yes, I lol'd as well. This topic got wayyyyyyyyy off track. YES you CAN break the speed of light, and NO you WOULD NOT instantly disintegrate NOR travel back in time. People sitting on earth watching you will never observe you break the speed of light, however you sitting on the ship accelerating at 1g constantly will break that speed like nothing ever happened,
...and when you flick on the lamp next to your bed on that ship in 141 days when you're traveling at the speed of light, the light from the lamp will move at the speed of light away from you like any lamp would on earth. (That's the speed of light^2 for you kids observing at home, however you'll never see it from your reference point, only the people on the ship will be able to see it. You'll end up just seeing a long stretched out line that is the ship).
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I suggest you read this article: https://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration
If you are thinking of things in a purely Newtonian way, the answer can be easily derived from the formula where time = the difference between the final velocity minus the initial velocity divided by acceleration. t=(vf-vi)/a. It would take about a year given that vf is about 300,000,000 m/s and 1g acceleration is about 10 m/s. Doing the math, you get about 30 million seconds, or about 350 days.
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The question isn't really valid in this universe. But to give an answer, you will NOT reach light speed, not from your reference frame or from any other. Sorry friends :( .
Here is what would actually happen:
According to Earth's reference frame, your acceleration will continually decrease as you approach the speed of light. They will never see you reach the speed of light. They will, however, see your energy/momentum approach infinity.
According to the ship's frame, you are at rest (lol). You are eternally moving at zero speed. Of course, that isn't what we want to hear, and it goes to show how careful we must be about defining these things. Even more so when we're talking about relativistic situations.
Now for what we actually want to discuss: your acceleration and speed with respect to the world around you. We must take relativistic effects into account. For example, the distance in front of you will CONTRACT the faster you go with respect to those objects. This means that while you traveled, say, 5000 light years according to Earth, from YOUR frame of reference you might have only moved 5 meters (because, again, the distance in front of you contracts-btw to get that to happen, you'd have to have reached about 99.9999999999999999999999999999999999999933 times the speed of light according to Earth's reference frame). What's your speed if you moved 5 meters in however many years? Not even close to c. But again, you're at rest in your frame anyway (and feel an artificial gravitational field). (note: it will look like 5 meters once you hit that speed, but don't lose track of the fact that the faster you go, the more contracted things will get, which means the slower you were, the LESS contracted things were; this post is a vast oversimplification).
In other words, according to the space ship, you'd be able to reach anywhere in the universe without ever hitting the speed of light. This is the important part of all that. As for how long that will take, well that's another topic (according to convertalot.com that would take about 16.5 years accelerating at 1g. Which means there is significant length contraction along the way, but not to the point that 5000 light years becomes 5 meters. Since we're accelerating, that 16.5 years is NOT constant. As time goes on, you'd be able to cover more "Earth measured" distance in less time per meter, but as for you, you'll never move a c with respect to the world around you. It will just keep getting more and more contracted in your direction of motion).
If anyone wants detailed proof about why this would happen, I challenge you to take some time to learn about the reasons why special relativity and general relativity were needed in the first place. It will require a lot of work on your part. If you don't want to do the work, don't be an arrogant fool like that John guy who just called all of relativity "garbage" simply because he couldn't understand it. We could pretty easily see why length contraction and time dilation have to be true, because it only requires algebra and at most basic calculus, but when dealing with acceleration things get a lot more complex. Fortunately the clock hypothesis is experimentally true, which helps with that, but I'm rambling off topic now.
Point is, if you believe in observable reality (and verified experiments), we know that due to characteristics of space and time, no one will ever see anyone accelerate to the speed of light, so the question is like asking what's the color of middle C.
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Now for what we actually want to discuss: your acceleration and speed with respect to the world around you. We must take relativistic effects into account. For example, the distance in front of you will CONTRACT the faster you go with respect to those objects. This means that while you traveled, say, 5000 light years according to Earth, from YOUR frame of reference you might have only moved 5 meters (because, again, the distance in front of you contracts-btw to get that to happen, you'd have to have reached about 99.9999999999999999999999999999999999999933 times the speed of light according to Earth's reference frame). What's your speed if you moved 5 meters in however many years? Not even close to c. But again, you're at rest in your frame anyway (and feel an artificial gravitational field). (note: it will look like 5 meters once you hit that speed, but don't lose track of the fact that the faster you go, the more contracted things will get, which means the slower you were, the LESS contracted things were; this post is a vast oversimplification).
I think, for all practical purposes, achieving faster-than-light travel, in the ship's frame of reference but according to Earth's distance metric, is sufficient. See:
It will never reach the speed of light. It would take an infinite amount of energy. This results from Einstein's theory of Special Relativity. However, from the perspective of anyone travelling on this craft, they will experience a contraction in the distances in their direction of travel. So although they can never get to 186,000 miles per second they can (theoretically), nonetheless, achieve a speed such that they can travel a distance that they may initially have measured (before accelerating) as 186,000 miles in less that 1 second as measured on their clocks. If I remember correctly this turns out to be the same time (to get to this speed) as would be calculated by Newtonian mechanics - I would need to check this with some maths to be sure. There are some unfortunate consequences of travelling this fast resulting from time dilation so that should return at some point you would find the earth you left having aged considerably compared to yourself.
If you could travel the diameter of the Milky Way, due to speed, time dilation, and length contraction, in under a lifetime, would you really split hairs and complain that it was actually a shorter distance and slower time while you were traveling there? No, you'd say that you traveled 100,000 light-years in under a lifetime, which, pragmatically speaking, is perfectly equivalent to faster-than-light travel.