Q1PE - College Physics (Urone)

Question

What is γ? (a) if v=\({\bf{0}}.{\bf{250c}}\)? (b) If v=\({\bf{0}}.{\bf{500c}}\)

 

Step-by-Step Solution

Answer

The Relativistic factor γ is

(a) γ=\(1.033\) 

(b) γ=\(1.155\)

Step 1 : Formula for Relativistic factor, γ

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

Step 2: To find the Relativistic factor, γ for v= \({\bf{0}}.{\bf{250}}\) c

(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. 

Step 3: To find the Relativistic factor, γ for v= \({\bf{0}}.{\bf{500}}\) c

(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.