Question

1. Give specific evidence from the song to back your claims. Feel free to reference the line numbers for convenience. Also, assume that the numbers have infinite significance.) Given the clues in the song from the time the travelers left to the time they retum, a) how much time had passed on Earth? b) how much time had passed on the ship? c) what is the Lorent from d) relative to the Earth, how fast were the brave souls" traveling? (Assume they traveled at a constant speed the entire time the turn around was instantaneous. They survived by a miracle.) 2. Assume that the travelers left Earth and that Earth scientist was telling the truth when he stated that the the distance between the Earth and the distant planet is constant. If the journey to the distant planet took half the time of the trip, then how far away in meters) was the planet from Earth, BG a) from Earth's point of view? b) from the ship's point of view (at its relativistic speeds)? 3. Making reasonable assumptions (which are to be stated), what would an Earth scientist measure as the total mass of the "the score brave souls inside" while they were in between planets? 4. If the ship sent a 600 nm wavelength signal to Earth for the entire trip, what frequency would the Earth receive a) when the ship was traveling away? b) when the ship was traveling back? 5. From the point of view of the people in the rocket, how far did they (the people in the rocket) travel? 6. If the rocket crashed into Earth in a perfectly inelastic collision without slowing down, assuming the Earth is at rest prior to collision, what is the speed of the Earth/rocket combination in the Earth's original reference frame? Assume the rocket has a mass of 150 x 10" kg when at rest relative to the Earth.

Answer 1

I am taking the answers 1 a) and 1 b) to be data.

Time in ship =100 y

time on earth =1 y

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$\Delta t_e=\gamma\Delta t_s\implies \gamma=\frac{\Delta t_e}{\Delta t_s}=\frac{100}{1}=100$

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$\newline\gamma=\frac{1}{\sqrt{1-(\frac{v}{c})^2}}\implies 10000=\frac{1}{1-\frac{v^2}{c^2}}\newline\newline\text{on rearranging, we get:}\newline\newline v=c\sqrt{\frac{9999}{10000}}\implies v=0.99994c$

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time = 50 years

speed = 0.9994c

distance =speed *time = 50*365*24*60*60*0.9994*3*10^8 = 4.727*10^17 m

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Lorentz contraction effect:

$l_s=\frac{l_e}{\gamma}=\frac{4.727\times 10^{17}}{100}=4.727\times 10^{15} m$

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masses get more massive by the formula:

$m=\frac{m_0}{\gamma}$

The rest mass m_0 is not given, but the above formula is to be applied.

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The Doppler effect for light:

$\newline \frac{1}{\lambda_{obs}}=\frac{1}{\lambda_{source}}\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}\newline\newline\text{ v is positive if rocket is travelling towards the earth.}\newline\newline\text{On substituting v = -0.9994c, we get:}\newline\newline \lambda_{obs}=34635.81 nm$

$\newline\text{On substituting v = +0.9994c, we get:}\newline\newline \lambda_{obs}=10.393 nm$

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Assuming that they felt the initial acceleration and the acceleration during the instantaneous turn, they must know they were the ones who were traveling.

Thus, they will just take the total distance as twice the Lorentz contracted length.

d_tot= 2*d =2*4.727*10^15 m = 9.454*10^15 m

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