Relativity Paradox: Explaining A Ship Moving at 198% Light Speed

[SmarterThanGod]

Lets say a ship, traveling 98% the speed of light approaches another ship traveling the opposite direction at the same speed. Both at a constant speed, so both have grounds for being at rest, at least from my understanding, but then wouldn't each record the other's speed to be 198% the speed of light? Relativity declares the universal "speed limit" to be light speed. I am probably missing something, so don't laugh at my naivety, but I can't figure it out.

 

[SmarterThanGod]

correction

sorry, my math was wrong there, it would be 196% light speed

 

[Janus]

Originally posted by SmarterThanGod
Lets say a ship, traveling 98% the speed of light approaches another ship traveling the opposite direction at the same speed. Both at a constant speed, so both have grounds for being at rest, at least from my understanding, but then wouldn't each record the other's speed to be 198% the speed of light? Relativity declares the universal "speed limit" to be light speed. I am probably missing something, so don't laugh at my naivety, but I can't figure it out.

The thing you are missing is that velocities don't add the way we thought they did under Newtonian physics.

To add two velocities such as in your example you need to use the formula:

[tex]w=\frac{u+v}{1+\frac{uv}{c^2}}[/tex]

In which case you get an answer of

0.99979596c

Since the advent of Relativity we have learned that this is the formula to use when adding velocities. It is just that in normal situations where u and v are small compared to c, the answer comes out to be so close to

[tex]w=u+v[/tex]

that we can use the second formula and get an answer that is within reasonable limits of error.

 

[SmarterThanGod]

thank you Janus, that makes sense to me now. I thought it might be something like that, but I've never seen that equation before. Thank you again.

 

[SmarterThanGod]

So, can anyone tell me WHY this is how it is? and not just because that's what einstein said. Are there actual physical reasons this equation exists? And how do we know this, if we've never accelerated 2 objects to 99% light speed?

 

[chroot]

Originally posted by SmarterThanGod
So, can anyone tell me WHY this is how it is? and not just because that's what einstein said. Are there actual physical reasons this equation exists? And how do we know this, if we've never accelerated 2 objects to 99% light speed?

It is a consequence of the fact that light always goes c in every frame of reference. The question of why light always goes c in every frame of reference doesn't really have an answer -- it just happens this universe operates that way.

- Warren

 

[suyver]

Originally posted by SmarterThanGod
And how do we know this, if we've never accelerated 2 objects to 99% light speed?

Sure we do!

In particle accelerators particles are routinely accelerated to velocities >0.99 c.

 

[SmarterThanGod]

Originally posted by suyver
Sure we do!

In particle accelerators particles are routinely accelerated to velocities >0.99 c.

I know this, but have 2 particles been accelerated towards each other, with some way of "viewing" one from the other?

Also, c always remains the same in all references, but c is not beign measured here, the speed of a ship in comparison with another ship is. Are you saying that the LIGHT from the ship can only appear to go this fast?

 

[Nereid]

SmaterThanGod wrote: I know this, but have 2 particles been accelerated towards each other, with some way of "viewing" one from the other?

"Collider" accelerators are exactly that, and the LEP (Large Electron Positron) collider is perhaps the most powerful, at least until the LHC (Large Hadron Collider) comes on stream at CERN.

One way to assess what one particle 'sees' of the other that it collides with is to measure the energies of the particles before they collide head on, and the total energy of the 'rubble' after the collision.

In short, each of the two particles 'sees' the other moving towards it at exactly the speed given by the equation in Janus' post.

 

[suyver]

Another good check is to use particles with quite short lifetimes, such as a muon (lifetime ~ 10 us). If you accelerate such a muon to velocities ~0.99 c then due to the time dilation the lifetime seems to increase in the lab-reference and you will get reasonable probabilities to still detect the muon after a few minutes.

 

[SmarterThanGod]

But according to the equation above, accelerating a particle to 99% light speed, and having it smash into a stationary particle, this collision will produce more energy that if 2 particles were accelerated towards each other at 99% light speed, or it would be really close, am i correct? This may be how it is, but it just doesn't seem logical to me

 

[Jess]

Originally posted by SmarterThanGod
But according to the equation above, accelerating a particle to 99% light speed, and having it smash into a stationary particle, this collision will produce more energy that if 2 particles were accelerated towards each other at 99% light speed, or it would be really close, am i correct? This may be how it is, but it just doesn't seem logical to me

The two-accelerated-particle case has a higher total energy (invariant mass/energy + kinetic energy) than the one-accelerated particle case, so if the two particles were to annihilate on contact, for example, the EM radiation produce in the first case would have higher energy (higher frequency) than in the second case.

That's not worded very well - I appear to have lost any ability I ever had to form sentences, but I hope you understand what I mean...

Jess

 

[Doc Al]

Originally posted by chroot
It is a consequence of the fact that light always goes c in every frame of reference. The question of why light always goes c in every frame of reference doesn't really have an answer -- it just happens this universe operates that way.

Let me just add that this is not just something weird about light---it's something weird about the structure of the universe. Anything traveling at the speed of light would be measured to have the same speed in any frame.

 

[SmarterThanGod]

Originally posted by Jess
The two-accelerated-particle case has a higher total energy (invariant mass/energy + kinetic energy) than the one-accelerated particle case, so if the two particles were to annihilate on contact, for example, the EM radiation produce in the first case would have higher energy (higher frequency) than in the second case.

That's not worded very well - I appear to have lost any ability I ever had to form sentences, but I hope you understand what I mean...

Jess

I think I understand it, but that doesn't explain there relative speeds beign almost identical, just that one has higher energy. How can accelerating one particle to 99% c with respect to earth, say, and another towards the first yeaild the same relative speed as a particle hurled at a stationary object? And how can the energy's be different if ones speed is the same?

 

[GijXiXj]

"And how can the energy's be different if ones speed is the same?"

Because E = mc^2, and m = m0/sqrt.(1 - v^2/c^2), so at relativistic speeds (more of) the energy you put in hoping to increase v instead increases the mass. But the energy is still there, and when it hits something else it's expended (or some of it is).

So if you took a (charged!) pea, and accelerated it in some stupendous accelerator to near light speed, it might eventually "weigh" (have the same mass as) a double-decker bus, but would actually be smaller than the original pea from the viewpoint of the experimenter! I don't need to tell you the effect of a a double-decker bus hitting something at near light speed (or two double-decker buses, ie: fast peas, hitting each other at near light speed ;-)

 

[ranyart]

Originally posted by GijXiXj
"And how can the energy's be different if ones speed is the same?"

Because E = mc^2, and m = m0/sqrt.(1 - v^2/c^2), so at relativistic speeds (more of) the energy you put in hoping to increase v instead increases the mass. But the energy is still there, and when it hits something else it's expended (or some of it is).

So if you took a (charged!) pea, and accelerated it in some stupendous accelerator to near light speed, it might eventually "weigh" (have the same mass as) a double-decker bus, but would actually be smaller than the original pea from the viewpoint of the experimenter! I don't need to tell you the effect of a a double-decker bus hitting something at near light speed (or two double-decker buses, ie: fast peas, hitting each other at near light speed ;-)

Obvious this would be a new Theory maybe called:The Big-Bus-Bang?

The question you then have to ask, what are the chances of two buses arriving at the same time?

 

[GijXiXj]

Obvious this would be a new Theory maybe called:The Big-Bus-Bang?

No, it would be the theory of the small pea crunch. Alternatively if the peas were energetic enough, they might create a brane, then that would most likely end the universe as we know it, and could be called the "Pea-Brane" Theory of the little crunch.

The question you then have to ask, what are the chances of two buses arriving at the same time?

I thought that's what they were supposed to do, rather than evenly separated in time (whatever that is), which would be more useful.

BTW, what the hell's a "Cave-Ranyart", and the "Moorglade"?

 

[ranyart]

Originally posted by GijXiXj


BTW, what the hell's a "Cave-Ranyart", and the "Moorglade"?

Ranyart on board the Moorglade.


http://www.iupui.edu/~philosop/spiritofolias.htm

It's just an insignificant collection of words?.. from one mind to another..every mind has a universe within:http://translate.google.com/transla...ias&start=10&hl=en&lr=&ie=UTF-8&oe=UTF-8&sa=N

 

[TheDonk]

One thing I don't understand is:
The fact that light always goes the same speed independant of the reference. How does this give an equation such as:
(u+v)/(1-u*v/c^2) which at low speeds is approx. u+v
instead of
(u+v)/(1-(u^2)*(v^2)/c^2) which at low speeds is approx. u+v

I'm pretty sure it has to do with the hypotenuse (sp) of some triangle light makes... but please explain... thanks

 

[chroot]

Originally posted by TheDonk

instead of
(u+v)/(1-(u^2)*(v^2)/c^2) which at low speeds is approx. u+v

The equation

[tex]u = \frac{v+u^\prime}{\sqrt{1+v^2 u^{\prime 2} / c^2}}[/tex]

would not be dimensionally consistent. Inside the square root, you have a dimensionless quantity (1) added to a quantity with units of velocity squared, which doesn't make sense. The total equation would not be dimensionally consistent.

- Warren

 

[that_guy]

Donk-

The units don't work out in the second equation so it really doesn't make any sense, unless you want to use c^4 instead of c^2.

 

[SmarterThanGod]

OK, here's a new situation let's say that a ship is going 50% light speed, and there is another ship approaching it going 51% light speed. To each other, they would appear to be going about 80% light speed. What if a third ship, which is said to be at rest, views their approach on each other. both ships would appear to be going their actual speeds, since the third ship is at rest. Would they appear to approach each other at 101% light speed? in other words, can 2 objects approach each other atr light speed according to another observer, or is there a law preventing this as well?

 

[Ambitwistor]

Originally posted by SmarterThanGod
[...] Would they appear to approach each other at 101% light speed?

Yes, they will, according to the third observer; that is allowed by relativity.

 

[NateTG]

Originally posted by SmarterThanGod
OK, here's a new situation let's say that a ship is going 50% light speed, and there is another ship approaching it going 51% light speed. To each other, they would appear to be going about 80% light speed. What if a third ship, which is said to be at rest, views their approach on each other. both ships would appear to be going their actual speeds, since the third ship is at rest. Would they appear to approach each other at 101% light speed? in other words, can 2 objects approach each other atr light speed according to another observer, or is there a law preventing this as well?

First off, there is 'at rest' in the sense that there is no single cannonical reference frame, and no, special relativity does not AFAIK prevent the appearance of things having relative velocities greater than C.

Similarly, the 'dot' on an ocilloscope or created by a laser can travel faster than 'c' because it's not acutally an object.

 

[TheDonk]

My computer busted hours after I posted my message with the equation:
(u+v)/(1-(u^2)(v^2)/(c^2))
and yes I did mean c^4.
Can anyone explain why the equation is not like this? Or basically where the real one came from because I always hear it's all based on the fact that light goes the same speed independant on reference but there must be at least a little more to it. Sorry for the slowness in response, my computer's still not fixed so I may be a day or 2 before I can respond again.