Question

At what speed will the mass of an electron be three times its rest mass

Answer 1

At what speed will the mass of an electron be three times its rest mass

Answer 2

The speed at which the mass of an electron will be three times its rest mass is $\frac{2\sqrt{2} }{3}c.$

Given,

Electron mass becomes three times its rest mass.

To find,

At what speed?

Solution,

Firstly, we know that, from the special theory of relativity, the relationship between the mass of a moving body (m) with its rest mass (m₀) is given as

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2} } } \hfill ...(1)$

where,

v = speed with which the body is moving,

c = speed of light.

As the mass is given to be 3 times its rest mass, we have,

m = 3m₀.

Substituting this value in (1),

$3m_0=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2} } }$

Rearranging and simplifying,

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

$1-\frac{v^2}{c^2} = (\frac{1}{3} )^2\\\implies \frac{v^2}{c^2} =1-\frac{1}{9}=\frac{8}{9}$

$\implies \frac{v}{c} =\sqrt{\frac{8}{9} }$

$\implies v = \frac{2\sqrt{2} }{3}c$

This is the required speed.

Therefore, the speed at which the mass of an electron will be three times its rest mass is $\frac{2\sqrt{2} }{3}c.$

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