Q2PE - College Physics (Urone)

Question

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

Step-by-Step Solution

Answer

The Relativistic factor γ is

(a) γ=\(1.005\) 

(b) γ=\(2.294\)

 

Step 1 : Formula for Relativistic factor, γ

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 \({10^8}\) \({\rm{m}}{{\rm{s}}^{{\rm{ - 1}}}}\)

Step 2: To find the Relativistic factor, γ for v= \({\bf{0}}.{\bf{100c}}\)

(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.

Step 3: To find the Relativistic factor, γ for v=0.900c

(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.