# Calculating the speed of a proton in the LHC at 7TeV without a calculator

There are several threads on this forum regarding the LHS at CERN however I am interested in a non-calc solution. I recently stumbled upon a multiple choice question about it while studying for a physics exam and am wondering if someone has an elegant way of solving it without a calculator. We are given a sheet with information such as proton mass ect.

The question states: According to a page on Wikipedia which describes the LHC particle accelerator (Large Hadron Collider), protons are accelerated to an energy of 7 TeV ≈ 1.12 μJ. It also states that the protons aquire a speed of about 0.999999991c. On CERN's own website, on the other hand, it has been stated that the protons get a speed of 0.9999c. What speed does a proton have with an energy of 7 TeV?

Here's how I would solve it using a calculator:

I cannot wrap my head around solving this with only a pen and paper especially considering the similarity of the answers. Could it be that they regard this as "presumed knowledge"?

• Hint: calculate $1-v/c$ and use Taylor series expansion for the square root. – Massimo Ortolano May 2 '19 at 16:57

$$\frac v c \equiv \beta = \sqrt{1-\frac 1 {\gamma^2}} \approx 1-\frac 1 2 \frac 1 {\gamma^2}$$

note:

$$\frac 1 {\gamma} = \frac {Mc^2} E = \frac{938}{7,000,000} \approx \frac 1 7 \frac 1 {1,000}$$

so

$$\frac 1 2 \frac 1 {\gamma^2}\approx \frac 1 2 \frac 1 {50} \frac 1 {1,000,000} = \frac 1 {10^8}$$

So $$\beta$$ is 8 nines, or

$$v = 0.99999999c$$

• I suggest you add a word statement regarding for the approximation in the first equation. The second equation line then shows that the approximation is justified. That would make for a more complete answer. – Bill N May 2 '19 at 17:48