Light travels to a certain star in 236 y. Anastronaut makes the trip in 12 y in

Question

         Light travels to a certain star in 236 y. Anastronaut makes the trip in 12 y in his frame ofreference.                     
a) How fast do we see him going (as a multiple ofc)? Answer in unitsof c.                                        
b) If he just signs the guest book and comes right back, inhow many years will he return as seen in the earth frame ofreference? Answer in units ofy.                                                                                   c).How much energy would it take toaccelerate a 1.3 Mg payload to this speed? The speed of light is 3× 108 m/s . Answer in units ofJ.                                                                                                           d). How many times greater than theclassical solution for the energy is the relativisticsolution?                e). How much mass must be annihilated inour matter-antimatter engines to achieve thisspeed if we assume100% efficiency? Answer in units of kg.          Light travels to a certain star in 236 y. Anastronaut makes the trip in 12 y in his frame ofreference.                     
a) How fast do we see him going (as a multiple ofc)? Answer in unitsof c.                                        
b) If he just signs the guest book and comes right back, inhow many years will he return as seen in the earth frame ofreference? Answer in units ofy.                                                                                   c).How much energy would it take toaccelerate a 1.3 Mg payload to this speed? The speed of light is 3× 108 m/s . Answer in units ofJ.                                                                                                           d). How many times greater than theclassical solution for the energy is the relativisticsolution?                e). How much mass must be annihilated inour matter-antimatter engines to achieve thisspeed if we assume100% efficiency? Answer in units of kg.

Explanation / Answer

(a) How fast do we him going (as a multiple of c)?Answer in units of c.
answer: let his speed in our frame be
v = c
gamma factor = = 1/(1-^2)
distance to star = 236 ly
in his frame, it is contracted to
236/ ly
v = c = (236/)c/12
= 236/(12) = 19.6667/
^2 = 386.7778(1-^2)
= (386.7778/387.7778) = .99871
v = .99871c
(b) If he just signs the guest book and comes right back, inhow many years will he return as seen in the earth frame ofreference? Answer in units of y.
= 19.6921
trip time in our frame = 2*12 years = 472.61 y
(c) How much energy would it take to accelerate a 1.3 Mgpayload to this speed? The speed of light is 3 x108 m/s. Answer in units of J.
mass m = m0
energy needed = m0 c^2 - m0 c^2 =m0 (-1)c^2 = 1300(19.6921-1)(3 * 10^8)^2= 2.18 x 10^21 J
(d) How many times greater than the classical solution for theenergy is the relavistic solution?
classically, the energy needed would be given by the KE formula
KE = (1/2)m0 v^2 = (.5)(1300)(.99871 * 3* 10^8)^2 = .06x 10^21 J
relativistic E/ KE = 37.5
(e) How much mass must by annihilated in our matter-antimatterengines to achieve this speed if we assume 100% efficiency? Answerin units of kg.
Mc^2 = m0 (-1)c^2
M = m0 (-1) = 24299.7 kg

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