Please show all work and formulas used I am trying to learn how to do these prob
ID: 1590868 • Letter: P
Question
Please show all work and formulas used I am trying to learn how to do these problems
A star is at rest relative to the Earth and at a distance of 1500 light-years. An astronaut wishes to travel from Earth to the star and age no more than 30 years during the entire round-trip journey. Assuming that the journey is made at constant speed and that the acceleration and deceleration intervals are very short compared with the rest of the journey, what speed is necessary for the trip? According to the astronaut, what is the distance from Earth to the star? According to someone on Earth, how long does it take the astronaut to make the round trip? It takes light 1500 years to travel from Earth to the star, but the astronaut makes the trip in 15 years. Does this mean that the astronaut travels faster than light? Explain your answer.Explanation / Answer
a)
t = t_0 / sqrt(1 - (v/c)^2)
t_0 = 30 years
d = 3000 Light Years
t = d/v
t = d/v = t_0/sqrt(1-v^2/c^2)
(1-(v^2/c^2))/v^2 = t_0^2/d^2
(c^2 - v^2)/v^2 = t_0^2 * c^2/d^2 (multiply both sides by c^2)
(c/v)^2 = 1 + (t_0*c/d)^2 ---- (note that lightyears = # * c, d = 3000 * c)
(v/c) = (1 + (t_0*c/d)^2)^-1/2
(v/c) = ( 1 + (30 * c / 3000 * c)^2)^-1/2 = 0.8574
v = 0.999 *c
b)
Because we are looking from Astronauts perspective, astronaut is not moving and universe is. Thus, L is unknown and L_0 = 1500 ly.
L = 1500 light years * sqrt(1 - 0.999^2) = 67.06 Light Years
c)
According to "stationary" earth, the astronaut travels 3000 c*years @ 0.999 c.
t = d/v = 3000 c*years / 0.999 c = 3003 years
d)
No, it does not mean that he travels faster than the speed of light as our relativistic equations above expose. In reality one can look at it as the distance that he travels actually shrinks - as described by answer b.
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