91Ó°ÊÓ

The relativistic factor,

\(\gamma = \frac{1}{{\sqrt {1 - {{\left( {\frac{v}{c}} \right)}^2}} }}\)

Where v is the velocity relative to an observer and c= \({\rm{3}}{\rm{.00}}\) X \({10^8}\) \({\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}\)

(a) Here given that the velocity relative to an observer is, v=0.250 c

\(\begin{align}\gamma &= \frac{1}{{\sqrt {1 - {{\left( {\frac{v}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{\left( {\frac{{0.250c}}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{(0.250)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - 0.0625} }}\\ &= 1.033\end{align}\)

Hence, the relativistic factor is 1.033.

(b)Here given that the velocity relative to an observer is, v=\(0.500\)c

\(\begin{align}\gamma &= \frac{1}{{\sqrt {1 - {{\left( {\frac{v}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{\left( {\frac{{0.500c}}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{(0.500)}^2}} }}\\ &= 1.155\end{align}\)

Hence, the relativistic factor is 1.155.