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## Homework Statement

In 2011, researchers at the OPERA experiment thought they had seen neutrinos with mass m and energy E = 28 GeV moving faster than light. The baseline between the source and the detector was 731 km, and the neutrinos seemed to arrive 60.7 ns early, compared to the maximum physical velocity c. (a) Using ##m^2c^4 = E^2 - p^2c^2 ##with p the momentum of the neutrinos, show that ##v = \frac{\delta E}{\delta p}= c\sqrt{1-\frac{m^2c^4}{E^2}}##. (b) Calculate the mass of such faster-than-light neutrinos in ##GeV/c^2##

## Homework Equations

##p=\gamma mv##

##m^2c^4=E^2-p^2c^2##

## The Attempt at a Solution

(a) ##m^2c^4=E^2-p^2c^2##

I've tried many calculations and never managed to find the correct result. I don't really understand the ##\frac{\delta E}{\delta p}##.

Am I right in trying to convert ##p## to ##mv## or is it ##p=\gamma mc##?

By using ##p=mv##, I manage to get :

$$m^2c^4=E^2-m^2v^2c^2$$

$$\frac{E^2-m^2c^4}{m^2c^2}=v^2$$

$$\frac{E^2}{m^2c^2}-c^2=v^2$$

$$v^2=c^2(\frac{E^2}{m^2c^2}-1)$$

$$v=c\sqrt{\frac{E^2}{m^2c^4}-1}$$

But this isn't using ##\frac{\delta E}{\delta p}## and the result isn't quite right. What am I missing?

(b) I'm kind of fuzzy on this - to me, photons travelling at the speed of light are massless, and if I apply the given formula to find the mass of a neutrino travelling faster than light...Well I hit a dead-end because it would be less than massless? Atleast that's the way I see it.

I've managed to work out the speed of the neutrino if it arrives 60.7ns earlier than light, which would be ##v=300007473 m.s^{-1}## assuming that ##c=3.0*10^8 m.s^{-1}##

All calculations including a gamma factor would not work to me, as ##1-\frac{v^2}{c^2}## would be less than 0

so the square root in the gamma factor would be less than 0... How would I then calculate the mass of the faster-than-light neutrino?

Thanks to everyone for the amazing help!