A person travels by a car at a speed of 180 km h ⁻¹ . By his wristwatch, the journey from station X to station Y takes exactly 10 hours . How much time is taken in the road frame by the car to go from station X to station Y ?

Step-by-step solution

The proper time measured by the wristwatch in the car is = 10 hours = 36000 s . The speed of the car is v = 180 km h ⁻¹ = 180 5 18 m s ⁻¹ = 50 m s ⁻¹ . The time measured in the road frame is t . According to the time dilation formula: t = 1 - v^2 c^2 Since v c , we can use the binomial approximation (1 - x)^ -1/2 1 + x 2 : t (1 + v^2 2c^2 ) The extra time taken in the road frame is: t - = v^2 2c^2 Substituting the values: t - = 36000 1 2 ( 50 3 10^8 )^2 t - = 18000 2500 9 10¹⁶ t - = 2000 2500 10¹⁶ t - = 5 10^6 10¹⁶