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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?

X3074861X
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3 Answers3

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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 $c$.

Alfred Centauri
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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.

Alfred Centauri
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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 :))

Howard Pautz
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