# Finding the speed of a relativistic particle

1. Apr 11, 2016

### alba

1. The problem statement, all variables and given/known data

This is not homework. I am trying to figure out a simple and quick way to find the speed that correspond to a given energy can you tell me if it is correct?
.
At LHC they had experiments that give protons 7 Tera eV:
we know that a proton is .938272 Gev , so 7*10^12/.938*10^9 = 7460 is the increase of masses.

Now if we reverse the formula for mass

2. Relevant equations

7460+1 = 1 / √1 -x^2

we get
(1-x^2) 7461^2 = 1
x2 = 74612/ 7461^2+1 x = 0.999999991

The result matches LHC info, is there any mistake?

If this is correct why isn't the reverse accepted any more and the current formula is
E = p 2c 2 + m 2c^4?

Thanks

2. Apr 11, 2016

### drvrm

looking at your estimates it appears that for large energies it may work.but its an approximation and yourx is nearly 1.
have you calculated for say energies 0,5 c etc,
if you are posing an alternative form pl. give in a formula form.relating energy and velocity

3. Apr 11, 2016

### alba

I applied the γ Lorenz formula. why doesn't it work at all energies?

Can you show me ho to find the KE (the energy we must supply) to en electron to reach 0.8c?
Or conversely the speed it reaches if we suply .511MeV of energy.
I read that thespeed is rougly 0.82 C can you tell me how to reach an accuracy of 5 digits, Please?

Thanks a lot

4. Apr 11, 2016

### drvrm

there are various approx. relations for different energy range
a comparative discussion has been done in the following ;
https://en.wikipedia.org/wiki/Energy–momentum_relation or on other sites
actually for a good range classical forms also works approx.
but the form used by general people is thesquare of energy related to p^2c^2 and square of rest mass energy ;as it is applicable to photons as well -particles with zero rest mass.
regarding accuracy of calculation -it depends on the no. of significant figures one needs in the range of values.
i wonder what is the problem in calculating electrons speed /energy?

5. Apr 11, 2016

### alba

That link is no help, why are there various approx ? Isn't the formula valid for all speeds?
the canonical formula says at .866 C there are 2 masses (1/√1-.75 (=.25)), minus one it means that it takes one electron mass .511 MeV of Ke to reach .866 C.
The point is that the Bertozzi experiment found the actual speed = .82
I'd like to learn to calculate with a certain accuracy energies required from .6 to .9 C.
If you can't be bothered to show me how it is done, please give me some useful links.
Thanks a lot, your help s highly appreciated