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An astronaut would like to travel to a star which is 5 light-years away, spend a year visiting friends, and then return to Earth, while only aging by 11/2 years during the trip. At what speed beta = v/c does he/she need to travel? The length of a space-ship is measured to be 1/3 of its proper length. What is the speed (in units of c) of the spaceship relative to the observer's frame? At what speed will the Galilean and Lorentz expressions for the length of an object in the direction of motion differ by (a) 0.1% (b) 1% (c) 10%?

Explanation / Answer

1) so 1/2 years traveling total so 1/4 years each way

t = d/v = (5c/gamma)/v = 5c* sqrt(1-v^2c^2)/v=1/2

5*sqrt(1-beta^2)/beta = 1/2

beta = 0.995

2) L = L0/gamma

1/3 = sqrt(1-beta^2)

beta=0.9428

3) a) L0 - L = 0.1E-2L

1- sqrt(1-beta^2) = 0.1E-2

beta = 0.0447

b) 1- sqrt(1-beta^2) = 0.01

beta = 0.141

c) 1- sqrt(1-beta^2) = 0.1

beta = 0.4359

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