Question

with what welocity should a particle move, so that it's mass increases by 25% •○ its rest mass?​

Answer 1

The particle should move with a velocity equal to 3c/5 so that its mass increases by 25 %.

Given:

The percentage increase in the mass of the particle = 25 %

To Find:

At what velocity should the particle move?

Solution:

→ The relationship between the mass (m) of the body and its velocity (v) is given as:

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

• where 'm₀' is the rest mass of the body.
• where 'c' is the speed of light.

→ In the given question:

The percentage increase in the mass of the particle = 25 %

∴ The mass of the body (m) = m₀ + (25/100)m₀ = m₀ + (1/4)m₀ = (5/4)m₀

                                           $\boldsymbol{ \because m=\frac{m_{0} }{\sqrt{1-\frac{v^{2} }{c^{2} } } } } }\\\\\boldsymbol{ \therefore \frac{5m_{0}}{4} =\frac{m_{0} }{\sqrt{1-\frac{v^{2} }{c^{2} } } } } }\\\\\boldsymbol{ \therefore \frac{5}{4} =\frac{1 }{\sqrt{1-\frac{v^{2} }{c^{2} } } } } }$

                                           $\boldsymbol{\therefore \sqrt{1-\frac{v^{2} }{c^{2} }}=\frac{4}{5} }\\\\\boldsymbol{\therefore1-\frac{v^{2} }{c^{2} }=\frac{16}{25} }\\\\\boldsymbol{\therefore \frac{v^{2} }{c^{2} }=1-\frac{16}{25} }\\\\\boldsymbol{\therefore \frac{v^{2} }{c^{2} }=\frac{9}{25} }\\\\\boldsymbol{\therefore \frac{v }{c }=\frac{3}{5} }\\\\\boldsymbol{\therefore v=3c/5 }$

→ The velocity 'v' of the body comes out to be equal to 3c/5.

Therefore the particle should move with a velocity equal to 3c/5 so that its mass increases by 25 %.

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