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How it Works »New Instructable »i want to know the derivation of relative velocity formula used by Albert Einstein to prove that nothing can exceed the speed of light. Is there any simple derivation? please help

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active| newest | oldestThese people did a very neat experiment to try and understand how light travels.

Your question about how and why light is our speed-limit,

will be much more understandable when you know about this experiment.

A

And do you know how to mark best_answer ?

More to follow your responses.

http://en.wikipedia.org/wiki/Special_relativity

and this is because it just covers the "special" case of non-accelerating reference frames. That is to say, all the reference frames examined in the chapter on Special Relativity are moving with constant velocity with respect to one another. The treatment for reference frames that are accelerating, is called General Relativity.

Anyway, Special Relativity starts with two postulates: (1) The laws of physics are the same in every inertial (non-accelerating) reference frame. (2) The speed of light is always observed to be constant.

It is the second postulate that is the tricky one, and the one that leads to funny results via numerous

thought experiments.Usually the first thought experiment involves this imaginary spaceship with a "light clock" flying by.

http://en.wikipedia.org/wiki/Time_dilation#Simple_inference_of_time_dilation_due_to_relative_velocity

This first experiment leads to the notion of "time dilation". To the observer outside the spaceship as it flies by, it appears that time inside the spaceship is moving more slowly.

And that's where the fun starts, is with time dilation. Then other thought experiments lead to "length contraction".

The requirement that momentum be conserved leads to this idea of relativistic mass, that is that somehow objects flying past at relativistic speeds seem to carry more momentum than you would expect from just the classical formula for momentum p =m*v, and what it looks like to the observer watching the thing fly by, is that the mass of that thing is changing as a its speed increases.

Also the formula for this mass increase turns out to be asymptotic as speed approaches the speed of light. The result being that being it would require an infinite amount of work (energy) to accelerate some thing with mass to a speed exactly equal to the speed of light.

Anyway, that's just sort of the hand waving explanation of how it works. You really should find a good Physics book and work through the thought experiments yourself, if you're not convinced.

Some reference may exist on the web. I tried looking in the usual places like Wikipedia, and hyperphysics.phy-astr.gsu.edu, but I still have not found a really good one that is clear, easy to follow, and has all the thought experiments.

Since energy and momentum form a covariant tensor (rank-1), written as (E,px,py,pz) with c=1, trying to interpret momentum in a Newtonian way is just wrong, and leads to all sorts of mathematical awkwardness.

This idea, I think, arose from trying to apply the naive form of Newton's force law relativistically. We commonly wright F = ma, notice that a = dv/dt, and then can say that, "since m is constant, F = m dv/dt = d(mv)/dt", and so force corresponds to change in momentum.

But starting from F=ma in that logic is putting the cart before the horse! What Newton actually said was (using modern language) that force

ischange in momentum.If you have a system with constant mass, then change in momentum is proportional to change in acceleration (in Newtonian physics). But if your system has a varying mass (like a rocket), then the relationship of force to mass and velocity is different, but the relationship to

momentumis not!F = dp/dtis always true, in any frame of reference; it is a basic law of physics.And that makes is perfectly relativistic! If you use F = dp/dt, then you get the right relationship even in SR, and you get the right connection between momentum and energy I stated above.

Perhaps one could say that the difference, relativistic mass minus rest mass, or (γ-1)*m0, is due to kinetic energy that

resembles massorlooks like mass, orfeels like mass, to the observer in the frame watching this thing go flying by. Maybe a statement like that could lead to a good hand waving explanation about the difficulty of accelerating an object to near the speed of light. You could say, well, the kinetic energy in this object "looks like mass" or "feels like mass". It contributes to the inertia, and thus the more kinetic energy you pour into it, the more difficult it is to make it move faster.The result of this being the experimental result that you guys with the particle accelerators can put huge amounts of energy and momentum into particles like protons and electrons, but you still can't get them to move from one side of the laboratory to the other any faster than a beam of light can.

;-)

Which reminds me: Thanks for linking to the updates on that story about those too-fast neutrinos. I mean the popular press made all kinds of noise about the initial story, but they have not been quick to follow up on these later reports of "Oh wait... Maybe it was a measurement error."

Who could travel much faster than light.

He set out one day

In a relative way

And came back the previous night

But this looks like a decent article breaking down the equation for you. But basically it comes down to this. The faster you travel the more energy you need to travel that fast. Since Einstein's equation shows us that Mass and Energy are 2 different way of measuring the same things (the same as using inches or miles to measure a distance). So the energy we are putting in is also adding more mass to the object being moved. So by the time you reach 90% of the speed of light the object has twice the mass it did before it started moving. Which in turn means it requires twice as much energy to keep that object moving. So you get diminishing returns as you approach the speed of light.

I think the article covers it a bit better then i can.

http://physicsworld.com/cws/article/news/48763

http://www.sciencenews.org/view/generic/id/338735/title/Loose_cable_blamed_for_speedy_neutrinos

http://www.sciencemag.org/cgi/content/full/335/6072/1027 (paywall)

That article is okay but it perpetuates the misconception of "relativistic mass." Mass is an intrinsic property of an object -- it measure how much matter (number of atoms of which kinds) the object has. The important kinematic (motion) quantities in relativity are energy (E) and momentum (p).

Those quantities are related by the same kind of equation as space and time: E

^{2}= p^{2}c^{2}+ m^{2}c^{4}(in physics, we usually set c=1 so we can avoid writing it down all the time). In this equation,mis a constant which tells you something about the object, no matter what frame of reference you're working in. In particular, thatmtells you how much inertia the object has -- if you apply a force, how much does the momentum change?Stories about the "impossible" are entertainment through imagination* I guess?

L

*As opposed to news.