Estimate mass of neutrino given distance & KE

hb1547

1. The problem statement, all variables and given/known data
A supernova [tex]1.54\times10^21 m[/tex] away sends out neutrinos, and a detector on Earth detects two, ten seconds apart. The first one (A) that comes has a kinetic energy of 30 MeV, the second (B) has a kinetic energy of 10 MeV.

Using this, I'm supposed to come up with an upper bound of the mass of a neutrino.

Approximations I can make:
- [tex]\gamma[/tex] is large
- The first particle travels at c

2. Relevant equations
- Lorentz transformations
- Length contraction/time dilation equations
- Relativistic kinetic energy
- Velocity formula

3. The attempt at a solution
Things I know:
- [tex]E_{A} = 3 E_{B}[/tex], so [tex]\gamma_{A} = 3 \gamma_{B}[/tex] too. This also implies that the length each particle feels it traveled (due to length contraction) is: [tex]L_{B} = 3 L_{A}[/tex].

I've tried using the approx. that the velocity of the second is c to come up with the time it took the particle to travel to reach Earth. Then, using t+10, I tried to get a velocity for the second particle. However, t ~ 5.13*10^12, so adding 10 seconds means that my computer's precision isn't high enough to give me useful answers, so I can't use that velocity to find Gamma, so I can't plug that into my energy equation and solve for the mass.

I've tried lots of random algebra hoping to get something simpler, but most of it just results in circular thinking that gets me nowhere.