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Very simple question, and frankly quite a silly one, but I'm currently writing a lecture for secondary school kids and I'd love to tell them how far the Large Hadron Collider could fire a proton.

The talk is actually about space, so what I'd really like to know is how would the velocity of the proton would compare to, for example, the necessary escape velocity for the proton to escape the Milky Way?

As the talk is mostly about energetics and kinematics I'd be keen to ignore the effect of the protons charge and just look at the kinetic energy being converted to gravitational potential.

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    $\begingroup$ "how would the escape velocity of the proton compare to, for example, the escape velocity from the Milky Way?" --- this doesn't make sense to me. What do you mean by this? In particular, what do you mean by the escape velocity of a proton. $\endgroup$
    – Brian Moths
    Commented Feb 24, 2014 at 22:05
  • $\begingroup$ Edited the question to helpfully be a bit clearer. Could also be phrased in terms of energy, what's the largest scale object the proton would have sufficient energy to escape from? $\endgroup$
    – zephyr
    Commented Feb 24, 2014 at 22:09
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    $\begingroup$ LHC protons are ultrarelativistic and should basically be able to escape anything that's not a black hole; as to the escape velocity of the Milky Way, according to Wikipedia it's less than that of the sun and thus well below the speed of light (though I cannot confirm right now as it's nightime where I'm sitting ;)). $\endgroup$
    – Christoph
    Commented Feb 24, 2014 at 22:10
  • $\begingroup$ That value is from the galactic radius, although it's true it has a few orders of magnitude to spare on ultra-relativistic speeds. It doesn't tell you however, the speed a body would need to be moving, starting at earth, to escape the milky way $\endgroup$
    – zephyr
    Commented Feb 24, 2014 at 22:13
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    $\begingroup$ If gravitational interactions in multiple star systems can fling a star clear out of the galaxy, into deep space, I don't see why a proton moving at .999999c or so would have any trouble escaping the galaxy. This is of course ignoring any drag due to collisions with other stuff, or attraction/repulsion due to electrical charge, or bending of the trajectory due to magnetic fields. Ignoring all that stuff, an accelerated proton should easily escape the galaxy, and probably even a Supergroup's gravity. $\endgroup$
    – Phil Perry
    Commented May 13, 2014 at 14:23

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I think I get what you're after. You're looking for a qualitative handle on just how fast those protons are slinging it...

You might try considering the speed differential between an LHC proton and a photon that is launched at the same time. As others have noted the proton is ultra-relativistic (its kinetic energy is waaay greater than its rest-mass), so working out an actual speed is a bit tricky. Luckily, someone already did it.

Using this value, the proton is moving at a speed of $(1 - 9\times10^{-9})c$ and the photon is moving at $c$. So the proton is 2.7 $ms^{-1}$ slower than the photon. The photon is getting away from the the proton at about the speed of a brisk walk.

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