Question

4. John, Jane, and Joe are twins and were all born 70 years ago (in Earth's frame of reference). John spent his life in the same city. Jane spent her life in a car moving at 30 m/s. Joe spent his life on a jet plane moving at 300 m/s. (Don't ask me how they did this.) How much younger are Jane and Joe than John? Calculate your answer in seconds. You will need to use the binomial approximation because your calculator is not accurate enough to calculate the difference between y and 1

Answer 1

The time contraction due to speed is given by the equation

$T = T_o\sqrt{1-\frac{v^2}{c^2}}$

Where T_o is the time measured in still frame

By Binomial expansion, Ignoring higher powers

$\sqrt{1-\frac{v^2}{c^2}} = (1-\frac{v^2}{c^2})^{1/2} = 1-\frac{v^2}{2c^2}$

Therefore age difference = T_o -T

= $\frac{v^2}{2c^2} T_o$

For Jane

T_o =70 years = 70*365* 86400 seconds, v = 30 m/s, c = 3*10⁸ m/s

Age difference

= $\frac{30^2}{2(3*10^8)^2} *70*365*86400 = 11.038 \mu s$

For Joe

T_o =70 years = 70*365* 86400 seconds, v = 300 m/s, c = 3*10⁸ m/s

Age difference

= $\frac{300^2}{2(3*10^8)^2} *70*365*86400 = 1.1038 ms$