The rest distance between patna and Delhi is 1000 km. A nonstop train travels at `360 km h ^(-1)`. (a) What is the distance between Patna and Delhi in the train frame ? (b) How much time elapses in the train frame between Patna and Delhi?

Explanation

Given ` L_0 = 1000 km = 10^(6) m ` ,
`v = 360 km / h `
` = 360 xx 5 / 18 = 100 m / sec`
(a) We know
` h' = h_0 ~(sqrt 1-v^(2)c^(2) )
` = 10^(6) (sqrt 1 - (100 / 3 xx 10^(8) ) ^(2))
` = 10^(6) (sqrt 1 - 10^(4) / ( 9xx 10^(16) ) `
` = 10^(6) (sqrt (9 xx 10^(16) - 10^(4)) / (9 xx 10^(16))) `
` = 10^(6) (sqrt 1 - 1 /9 xx 10^(-12))`
Solving we get Change in length ` = 56 nm`
(b) ` Delta t = Delta L / v = 56 nm / 100 m = 0.56 ns ` .