91Ó°ÊÓ

Given that, A proton having total energy 500 GeV .

We need to find it's velocity in the fraction of c.

Total energy is $E=500\mathrm{GeV}=500×{10}^9\mathrm{eV}=500×{10}^9×1.6×{10}^{-19}\mathrm{J}$.

Mass of the proton is 1.67 × 10^-27 kg.

So if V is the velocity of the proton then we have

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

$⇒500×{10}^9×1.6×{10}^{-19}=\frac{1.67×{10}^{-27}×{\left(3×{10}^8\right)}^2}{\sqrt{1-v^2/c^2}}$

$⇒V=Â\pm 0.999998c$.

So, The velocity of proton is$0.999998c$.

Given that, An electron having total enegy 2.0 GeV .

We need to find it's velocity in the fraction of c.

The energy of the electron is

$E=2.0\mathrm{GeV}=2.0×{10}^9×1.6×{10}^{-19}\mathrm{J}$.

And the mass of the electron is $9.10×{10}^{-31}kg$.

Hence we have

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

$⇒2.0×{10}^9×1.6×{10}^{-19}=\frac{9.2×{10}^{-27}×{\left(3×{20}^8\right)}^2}{\sqrt{1-v^2/c^2}}$

$⇒v=Â\pm 0.99999997c$

So the velocity of electron will be $0.99999997c$.