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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 [tex]E={\gamma}m_0c^2=\frac{m_0c^2}{\sqrt{(1-\frac{v^2}{c^2}}}[/tex]

for each particle and got

[tex]v_1=c\sqrt{(1-10^{-12})}[/tex] and [tex]v_2=c\sqrt{(1-\frac{10^{-13}}{9})}[/tex]

and therefore the difference in time is

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

When I plugged it into maple, I got [tex]{\Delta}t= 1000 s[/tex]. 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.

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 [tex]E={\gamma}m_0c^2=\frac{m_0c^2}{\sqrt{(1-\frac{v^2}{c^2}}}[/tex]

for each particle and got

[tex]v_1=c\sqrt{(1-10^{-12})}[/tex] and [tex]v_2=c\sqrt{(1-\frac{10^{-13}}{9})}[/tex]

and therefore the difference in time is

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

When I plugged it into maple, I got [tex]{\Delta}t= 1000 s[/tex]. 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.

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