Relativistic Energy: Calculating Time Difference for Neutrinos from Supernova

[Warr]

Hey I have a problem concerning relativistic energy

One neutrino has an energy of 10 MeV and a rest mass of 10 eV/c^2. Another neutrino has an energy of 30 MeV and a rest mass of 10 eV/c^2.

Calculate the difference in time that the two particles arrive at Earth if they are emitted from a supernova 150,000 lightyears away.

Unless I am doing the calculation wrong, the difference is almost negligable.

I used $$E={\gamma}m_0c^2=\frac{m_0c^2}{\sqrt{(1-\frac{v^2}{c^2}}}$$

for each particle and got

$$v_1=c\sqrt{(1-10^{-12})}$$ and $$v_2=c\sqrt{(1-\frac{10^{-13}}{9})}$$

and therefore the difference in time is

$${\Delta}t=\frac{d}{c}\left(\frac{1}{\sqrt{(1-10^{-12})}}-\frac{1}{\sqrt{(1-\frac{10^{-13}}{9})}}\right)$$

When I plugged it into maple, I got $${\Delta}t= 1000 s$$. But there is no way I would have gotten this through a calculator (would have rounded it to 0 since each of the velocities would round to c. So either I am doing it wrong, or I need to find a way to simplify the expression so that I don't require a computer program to get the answer.

 

[OlderDan]

I don't think those velocities are correct, and I think I can do the problem on my calculator. Check your work.

 

[Warr]

I checked them, and found the exponents to be off by 1 each, so I changed them to -12 and -13. Other than that, I still can't do it with my calculator. If I try to plug in the final expression, I simply get 0. Also, I plugged the expressions I got for the velocities back into my original expression for E, and got 10^7 eV and 3*10^7 eV respectivly, exactly what it should be. Also, I checked my maple calculationa and found it I forgot to divide by c, so I got $${\Delta}t=3.3*10^{-6}s$$.

 

[OlderDan]

I checked them, and found the exponents to be off by 1 each, so I changed them to -12 and -13. Other than that, I still can't do it with my calculator. If I try to plug in the final expression, I simply get 0. Also, I plugged the expressions I got for the velocities back into my original expression for E, and got 10^7 eV and 3*10^7 eV respectivly, exactly what it should be. So the only thing that I can assume is that I am doing the whole problem wrong.

I noted the exponents problem, but you were right about the calculator. I cannot do it on mine either. I don't think your approach is wrong.

OK. You should be able to do this using the binomial theoem. Since the important thing is the difference between the velocities, not the velocities themselves, expand the velocity equations and take the difference. Then calculate the time difference.

http://hyperphysics.phy-astr.gsu.edu/Hbase/alg3.html#bt

Only the first and second terms matter, and the first terms go away when you take the difference.

I think your last result is way off

 

[Warr]

I ended up getting about 2.37 s using the expansion (Also found that the second exponent should actually be -14). Does this seem right?

 

[OlderDan]

I ended up getting about 2.37 s using the expansion (Also found that the second exponent should actually be -14). Does this seem right?

I got 2.1 seconds. I think you're still a bit off on the velocity calculations

(10eV/10Mev)^2=10^(-12)

(10eV/30Mev)^2=(1/9)(10eV/10Mev)^2=(1/9)10^(-12)=1.111111111x10^(-13)

 

[Warr]

I recalculated and got that. I was plugging in 1 eV for the second particle instead of 10eV. Thanks for your help.