Light travels to a certain star in 159 y. An astronaut makes thetrip in 18 y in

Question

Light travels to a certain star in 159 y. An astronaut makes thetrip in 18 y in his frame of reference. (a) How fast do we him going (as a multiple of c)?Answer in units of c. (b) If he just signs the guest book and comes right back, in howmany years will he return as seen in the earth frame of reference?Answer in units of y. (c) How much energy would it take to accelerate a 1.6 Mg payload tothis speed? The speed of light is 3 x 108 m/s.Answer in units of J. (d) How many times greater than the classical solution for theenergy is the relavistic solution? (e) How much mass must by annihilated in our matter-antimatterengines to achieve this speed if we assume 100% efficiency? Answerin units of kg. (a) How fast do we him going (as a multiple of c)?Answer in units of c. (b) If he just signs the guest book and comes right back, in howmany years will he return as seen in the earth frame of reference?Answer in units of y. (c) How much energy would it take to accelerate a 1.6 Mg payload tothis speed? The speed of light is 3 x 108 m/s.Answer in units of J. (d) How many times greater than the classical solution for theenergy is the relavistic solution? (e) How much mass must by annihilated in our matter-antimatterengines to achieve this speed if we assume 100% efficiency? Answerin 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 = 159 ly
in his frame, it is contracted to 159/ ly
v = c = (159/)c/18
= 159/(18) = 8.83333/
^2 = 78.02778(1-^2)
= (78.02778/79.02778) = .99365
v = c = .99365c
(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.
= 8.88978
trip time in our frame = 2*18 years = 320.03 y
(c) How much energy would it take to accelerate a 1.6 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 = 1600(8.88978-1)(3 * 10^8)^2 = 1.14 x10^21 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)(1600)(.99365 * 3* 10^8)^2 = .071x 10^21 J
relativistic E/ KE = 16.06
(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) = 1600(8.88978 - 1) = 12623.6 kg

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