Question

Work out the following a) Find β = v/c for a person walking at 3 mile/hr and a truck moving at 65 mile/hr. (b) Find γ-1, where γ¡s the relativistic factor γ 1/ VI-B" for the velocities in (a). You should expand γusing the small velocity approximation to get the best results

Answer 1

Explanation:

It is given that,

Speed of the person, v = 3 mile/hr = 1.34 m/s

Speed of the truck, v' = 65 mile/hr = 29.05 m/s

(a) Since, $\beta =\dfrac{v}{c}$

For the person, $\beta =\dfrac{1.34}{3\times 10^8}=4.47\times 10^{-9}$    

For the truck, $\beta =\dfrac{29.05}{3\times 10^8}=9.68\times 10^{-8}$

(b) The relativistic factor is given by :

$\gamma=\dfrac{1}{\sqrt{1-\beta^2}}$

For very small velocity, $\beta$

$\gamma=(1-\beta ^2)^{-1/2}\approx 1+\dfrac{1}{2}\beta ^2$

$\gamma-1=\dfrac{\beta^2}{2}$

For the person :

$\gamma-1=\dfrac{(4.47\times 10^{-9})^2}{2}=9.99\times 10^{-18}$

For the person :

$\gamma-1=\dfrac{(9.68\times 10^{-8})^2}{2}=4.68\times 10^{-15}$

Hence, this is the required solution.