# Speed difference between protons & heavy-ions in LHC?

1. Jun 15, 2015

### Michel_vdg

On the wiki-pages it says that in the LHC protons travel 3m/s slower than c.

I was curious what it would be for heavy-ions; is it maybe 15m/s because their energy is ~5 x lower (7 vs. 2.76 Tev) ?

And a small question on the side, does Gravitation assistance have a bigger impact on speeding up Heavy-ions- vs. Proton-cosmic rays since they are more massive, or is gravitation assistance something negligible for particles moving so close to the speed of light?

--
protons have an energy of 7 TeV ... total collision energy of 14 TeV ... move at about 0.999999991 c, or about 3 meters per second slower than the speed of light (c)

heavy-ion (Pb-Pb) collisions at a center of mass energy of 2.76 TeV per nucleon pair .... accelerated to 99.9% of the speed of light
https://en.wikipedia.org/wiki/ALICE:_A_Large_Ion_Collider_Experiment
--

2. Jun 15, 2015

### e.bar.goum

0.999 *c = 2.995 * 10^8 m/s

vs

0.999999991*c = 2.99792455×10^8 m/s

ETA: Another way to do it: $E_k = (\gamma -1 ) m c^2$. So if you know the energy, you know the velocity. But you can already see that 3 times less energy doesn't lead to "three times fewer c-v (m/s)"

Last edited: Jun 15, 2015
3. Jun 15, 2015

### Michel_vdg

Alright, thanks.

mh, I just thought that the 99,9% for the ions was rounded off, cause on the wiki page of RHIC it says (but the write typically and p's are included in the list):

"To date the types of particle combinations explored at RHIC are p + p, d + Au, h + Au, Cu + Cu, Cu + Au, Au + Au and U + U. The projectiles typically travel at a speed of 99.995% of the speed of light."
https://en.wikipedia.org/wiki/Relativistic_Heavy_Ion_Collider

What does $\gamma$ production mean?

4. Jun 15, 2015

### e.bar.goum

$\gamma$ in this context is the relativistic quantity $\frac{1}{\sqrt{1-v^2/c^2}}$

If you're worried about rounding errors, you can do it the long way. The energy of the beam is 2.76 TeV per nucleon pair, so the kinetic energy of one lead ion will be 2.76 TeV * 208/2 = 287 TeV.

Now, if you know that $E = (\frac{1}{\sqrt{1-v^2/c^2}} -1 )m*c^2$, and you know the Energy, $E$ and you can google the rest mass of lead, it's just a matter of solving for $v$ and substituting in $E$, $m$ and $c$. I'll let you do that.

5. Jun 15, 2015

### Staff: Mentor

99.9% was a bad rounding error, fixed now.
The difference to the speed of light is more than 15 m/s, but still in the speed range of cars and trains.

6. Jun 15, 2015

### Michel_vdg

Thanks ... but solving mfb's reply is slightly faster : )

So it says now 99.9999% on the Alice wiki-page, this gives:

c = 299.792.458 m/s

Protons: 99.9999991% c = 0.999999991 c = 299.792.455
299.792.458 - 299.792.455 = 3 m/s slower

Heavy-ions: 99.9999% c = 0.999999 c = 299.792.158
299.792.458 - 299.792.158 = 300 m/s slower

btw the fastest train is a Maglev (magnetic levitation) train with 603 km/h = 167,5 m/s

7. Jun 15, 2015

### Staff: Mentor

No, it says "more than 99.9999%". That way it is true for both 2013 and 2015. Without a calculation you can just tell that the difference is below 300 m/s.