The LHC - Proton Speed Limit [duplicate]

I'm originally from stack overflow, so this is my first foray into this particular community.

I've read in numerous places that the Large Hadron Collider is capable of accelerating protons at 0.999999991 c, which mathematically works out to being 3 metres per second slower than the speed of light. That seems so incredibly close to the speed of light, that it's hard for me to understand why we can't quite get all the way there.

My Main Question :

What is preventing the LHC from achieving 1 c, in terms of how fast it can propel these protons?

Side Question :

If we were able to accelerate the protons to the speed of light, what would the results of their collisions be?

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marked as duplicate by akhmeteli, John Rennie, Emilio Pisanty, Waffle's Crazy Peanut, Michael BrownOct 21 '13 at 2:35

The "speed limit" of $c$ is asymptotic. That is, the more energy you add to the protons, the closer they will approach $c$, but it will never actually reach $c$ completely. (Equivalently, we say that it takes an infinite amount of energy to accelerate a massive object to $c$) –  Dmitry Brant Oct 18 '13 at 20:36
I searched the site before asking, and I couldn't find the question anywhere, but wow this place is unforgiving. I'll just go back to SO now... –  X3074861X Oct 18 '13 at 20:44
Was it something I said? –  Dmitry Brant Oct 18 '13 at 20:46
@X3074861X - it's like SE is today for noobs - just hang around awhile, the old fuddy-duddy physicists will knock you about a while, then become your pals :-P –  Howard Pautz Oct 18 '13 at 20:53
@X3074861X - probably downvoted because it's one of those questions every physics person knows the answer to off the cuff ... like "do C++ and Java have for loops?" for us coders ;-) –  Howard Pautz Oct 18 '13 at 21:06

Reaching the speed of light requires infinite energy. Let a proton of mass $m_p$ have a velocity $v$. Then by the energy-mass equivalence:

$$E = \frac{m_pc^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$

Which goes to infinity as $v$ approaches $c$. Since you can't supply infinite energy to the proton, reaching $c$ is impossible.

You can get close to $c$ as the LHC does but you will never ever reach $c$.

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What is preventing the LHC from achieving 1 c, in terms of how fast it can propel these protons?

The nature of the spacetime within which we exist.

There are several ways to look at this and here's just one. According to Special Relativity, the momentum of a proton with velocity $\vec v$ is given by:

$$\vec p = \dfrac{m_p \vec v}{\sqrt{1- \frac{v^2}{c^2}}}$$

Carefully note the denominator. As the speed v of the proton gets close to c, the denominator gets close to zero so the momentum increases without bound. What seems to happen is that, as the speed closes in on c, the applied force increases the momentum without significantly increasing the velocity.

Also note that, mathematically, the momentum doesn't exist when $v = c$ since division by zero is undefined.

So, the speed can become arbitrarily close to c but it can never equal c.

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X307 (I'm also a StackOverflow expat :) To get to a full 1 c, you would need an infinite amount of energy. No ones gas tank or budget's that big :))

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