Question

In your frame, a person is moving in the positive x-direction at a speed of 40,000 km / s. Suppose that a streetlamp located at x = 4 km turns on just as the person passes it by. The proper time interval between when the person is located at x = 0 km and when the streetlamp turns on is

Answer 1

Answer:

proper time taken by the person is 9.911 × 10⁻⁵ s

Explanation:

speed of the person  in x- direction = 40,000 km/s

                                                           =  40,000 × 10³ m/s

                                                           =  4 × 10⁷ m/s

when the person just passes the street lamp is switched on which is at x =4 km

Lorentz factor = $\gamma = \dfrac{1}{\sqrt{1-\dfrac{r^2}{c^2}}}$

                                         = $\dfrac{1}{\sqrt{1-\dfrac{(4 \times 10^7)^2}{(3 \times 10^8)2}}}$

                                         = 1.009

time taken in your frame of reference,t =$\dfrac{D}{v}$

                                                                 =$\dfrac{4}{40000} = 10 ^{-4}s$

proper time  =$t_0 = \dfrac{t}{\gamma}=\dfrac{10^{-4}}{1.009}= 9.911 \times 10^{-5} s$

hence, proper time taken by the person is 9.911 × 10⁻⁵ s