What speed does a proton have with the energy 7 TeV?

So this is a question from the 2014 admission test for Chalmers University of Technology, Gothenburg.

To clarify some rules (cited and translated from the test):

Calculators and physics formularies not allowed.

On the last page there is a list of physical constants, etc. which may be useful. For questions where numerical answers are sought it is sufficient with one or two significant digits, depending on the number of significant digits in the given question. Do not forget units in your answers.

The page with "useful constants" provides you with the information as seen below. So now to the question, which follows:

According to the LHC article on Wikipedia, protons are accelerated such that their mass energy is 7 TeV ≈ 1.12 µJ. Wikipedia states that the protons then have a speed of about 0.999999991c. However, CERN states own their own website that the relative speed of the protons is 0.9999c.

What speed does a proton with the mass energy 7 TeV have?

A. 0.9 c

B. 0.9999 c

C. 0.999999991 c

D. 0.9999999999991 c

How am I supposed to solve this without a calculator? For all these values, $\gamma$ is high as $v$ is close to $c$. As I have no access to a calculator according to the rules, I cannot calculate a numerical value of $\gamma$, at least not in a way I can tell the alternatives apart.

• Incidentally, "mass energy" is the wrong term to use here. The mass energy of a proton is always 938 MeV; it's the total energy, or total relativistic energy, that goes up to 7 TeV. The kinetic energy is the total energy minus the mass energy, and in this case that is also 7 TeV to a few significant figures. – David Z Oct 1 '15 at 13:34
• An exam question that cites Wikipedia; Jimbo Wales will be pleased to see this! – Count Iblis Oct 1 '15 at 16:12

I'll revise my answer, since my last answer was too specific and was deleted

You can start with an equation relating relativistic kinetic energy to velocity:

$E_k = m_0c^2\left[ \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right]$

For these types of problems where you can't use a calculator, you need to be able to estimate the order of magnitude of the quantities involved. Since you see that the answers each differ by several orders of magnitude, estimating each quantity involved to powers of 10 will generally suffice. You will want to find a way to write the quantity that you are interest in, in this case $v/c$, as something like

$\frac{v}{c} = \left(1-x\right)^n$

where $x$ is a quantity much less than 1, and $n$ is an arbitrary exponent based on the problem at hand. You can do a binomial expansion on this expression to say

$\frac{v}{c} \sim 1-nx$

Now, if you can estimate the value of $nx$ to within a factor of ten using the quantities you were given, you can solve this problem without a calculator.

• They deleted it because it were too specific? It answered my question perfectly. – Markus Klyver Oct 1 '15 at 14:57
• I am not supposed to answer these types of questions so directly, since it is either an exam question or a homework problem. For these types of questions, only conceptual help is allowed. – tmwilson26 Oct 1 '15 at 15:00
• @tmwilson26 Well, this isn't a homework question and the answer is publicly out. – Madde Anerson Oct 5 '15 at 9:45