Question

If the total energy of a particle is exactly thrice its rest energy, the velocity of the particle is:

Answer 1

Answer:

The velocity of the particle is $\frac{2\sqrt{2} }{3} c$

Explanation:

Here we have been given the total energy of the particle is exactly thrice of its rest energy. We have to find the velocity of the particle. Now as we know that the energy of the particle is given by the formula;

$E$ = $m c^{2}$

Here E is the energy of the particle, m is the mass of the particle and c is the speed of light in a vacuum.

Now it has been mentioned that the energy or total energy (E) is thrice its rest energy, therefore:

E = 3 × $m_{0} c^{2}$

here $m_{0}$ is the rest mass of the particle.

Now we have:

∴ $3m_{0} c^{2}= m c^{2}$

Now as we know that mass m of a particle is given by the following:

m = $m_{0} / \sqrt{1 - \frac{v^{2} }{c^{2} } }$

⇒ $3m_{0} c^{2} = m_{0} c^{2}/ \sqrt{1 - \frac{v^{2} }{c^{2} } }$

⇒ $\sqrt{1 - \frac{v^{2} }{c^{2} } } = \frac{1}{3}$

Squaring on both sides we get,

⇒ $1 - \frac{v^{2} }{c^{2} } = \frac{1}{9}$

⇒ $\frac{v^{2} }{c^{2} } = 1 - \frac{1}{9}$

⇒$\frac{v^{2} }{c^{2} } =\frac{8}{9}$

⇒ v = $\frac{2\sqrt{2} }{3} c$

Therefore the velocity of the particle is $\frac{2\sqrt{2} }{3} c$