Captain Kirk wants to go to a distant star which is five lightyears away. Calcul

Question

Captain Kirk wants to go to a distant star which is five lightyears away.   Calculate the velocity of his rocketrelative to the Earth so that the time as measured by Spock’sclock is one Earth year. What will be the time taken asobserved by the mission control on Earth?

Explanation / Answer

Captain Kirkwants to go to a distant star which is five light yearsaway.   Calculate the velocity of his rocket relative tothe Earth so that the time as measured by Spock’s clock isone Earth year. What will be the time taken as observed by themission control on Earth? This problem is easier tosolve if standard MKS units (e.g., "m", "m/sec", etc.) are used. Thus, values given in non-MKS units like "light-years"are first converted to MKS in the followingsolution: {Distance To Star, Earth Frame (given)} = (d_e) = 5.0 light-years =      = (5.0)*{Distance Light Travels In1.0 Year}      = (5.0)*{Speed OfLight (m/sec)}*{Number Of Seconds In 1.0 Year}      = (5.0)*(2.998e+8 m/sec)*(1.0 yr)*(365 days/yr)*(24hrs/day)*(3600 sec/hr)      = 4.7272464e+16 m {Distance To Star, Rocket Frame} = (d_r) =      = (d_e)*sqrt(1 - (v/c)^2)        (v/c)*(3.1536e+7) = (4.7272464e+16)*sqrt(1 - (v/c)^2)/(2.998e+8) ---->   (v/c) = (4.7272464e+16)*sqrt(1 -(v/c)^2)/((2.998e+8)*(3.1536e+7)) ---->    (v/c) = 5*sqrt(1 - (v/c)^2) ---->    (v/c) = 0.9806 ---->    {RocketVelocity, Earth Frame} = v = (0.9806)*c = 2.93984e+8 m/sec {Travel Time To Star,Earth Frame} = (t_e) =      = (t_r)/sqrt(1 - (v/c)^2)    

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