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The velocity of a particle at which the kinetic energy is eqyal to its rest mass energy is
(A) \({\left(\frac{{{3}{c}}}{{{2}}}\right)}\)
(B) \({3}\frac{{{c}}}{{\sqrt{{{2}}}}}\)
(C) \(\frac{{{\left({3}{c}\right)}^{{{1}/{2}}}}}{{{2}}}\)
(D) \(\frac{{{c}\sqrt{{{3}}}}}{{{2}}}\)

Answer

Answer: D
The relativistic K.E. of a particle of rest mass \({m}_{{{0}}}\) is given by
\({K}={\left({m}-{m}_{{{0}}}\right)}{c}^{{{2}}},\) where\({m}=\frac{{{m}_{{{0}}}}}{{\sqrt{{{1}-\upsilon^{{{2}}}/{c}^{{{2}}}}}}}\)
Here, \({m}\) is mass of particle moving with velocity \(\upsilon\).
As \({K}.{E}.=\) rest mass energy
\(\therefore{\left({m}-{m}_{{{0}}}\right)}{c}^{{{2}}}={m}_{{{0}}}{c}^{{{2}}}\) or \({m}{c}^{{{2}}}={2}{m}_{{{0}}}{c}^{{{2}}}\)
\(\frac{{{m}_{{{0}}}}}{{\sqrt{{{1}-\upsilon^{{{2}}}/{c}^{{{2}}}}}}}={2}{m}_{{{0}}}\)
\({1}-\upsilon^{{{2}}}/{c}^{{{2}}}=\frac{{{1}}}{{{4}}}\)
\(\upsilon^{{{2}}}/{c}^{{{2}}}={3}/{4},\upsilon=\frac{{\sqrt{{{3}}}}}{{{2}}}{c}\)
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