What is γ? (a) if v=\({\bf{0}}.{\bf{100c}}\)? (b) If v=\({\bf{0}}.{\bf{900c}}\)

The Relativistic factor

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

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=\(0.100\)c

\(\begin{align}\gamma &= \frac{1}{{\sqrt {1 - {{\left( {\frac{v}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{\left( {\frac{{0.100c}}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{\left( {\frac{{0.100c}}{c}} \right)}^2}} }}\\ &= 1.005\end{align}\)

Hence, the relativistic factor is 1.005.

(b)

Here given that the velocity relative to an observer is,\(v = 0.900\)c

\(\begin{align}\gamma &= \frac{1}{{\sqrt {1 - {{\left( {\frac{v}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{\left( {\frac{{0.900c}}{c}} \right)}^2}} }}\\ &= \frac{1}{{\sqrt {1 - {{\left( {\frac{{0.900c}}{c}} \right)}^2}} }}\\ &= 2.294\end{align}\)

Hence, the relativistic factor is 2.294.