Suppose you are planning a trip in which a spacecraft is totravel at a constant velocity for exactly six months, as measuredby a clock on board the spacecraft, and then return home at thesame speed. upon return, the people on earth will have advancedexactly 110 years into the future. according to special relativity, how fast must you travel? (give your answer with at least sixsignificant digits as a multiple of c.)

0.999959 c

Explanation:  

According to the special relativity theory, time is measured differently by two observers moving one relative another, according to the Lorentz Transform Equation, as follows:

t = t’ / t=t^'/√(1-(v)2/c2 )

where t= time for the moving observer (relative to the spacecraft, fixed on Earth) = 110 years.

t’= time for the observer at rest respect from spacecraft = 1 year

v= spacecraft constant speed

c= speed of light

Solving for v, with a six decimals precision as a multiple of c, we get:

v = 0.999959 c

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west