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

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

Step-by-step 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 = \frac{1}{\sqrt{1-(r^2)/(c^2)}}`

=
`\frac{1}{\sqrt{1-((4 * 10^7)^2)/((3 * 10^8)2)}}`

= 1.009

time taken in your frame of reference,t =
`(D)/(v)`

=
`(4)/(40000) = 10 ^(-4)s`

proper time =
`t_0 = (t)/(\gamma)=(10^(-4))/(1.009)= 9.911 * 10^(-5) s`

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