1. The problem statement, all variables and given/known data The last of the human race are leaving the earth, after a total nuclear destruction, to reach the only known planet suitable for lives, 2 million light years away from earth. They are traveling on the spaceship ARK, capable of close to speed of light. There is only enough food and energy to last 10 years for those on board. a). How fast must the ship travel so that the human race survive? b) If you made any assumption in part a), state it and justify it using your final results. c) In the eyes of the ship crew, how far did they travel when they reach the planet? d) In your calculation, which reference frame is the “proper frame”? Just posting my work to see if anybody can spot any errors with it. Thank you all in advance! 2. Relevant equations Δt = Δt_p*γ L = (L_p)/γ γ = 1/sqrt(1-β^2 ) β = v/c c = 3 * 10^8 m/s L_p = 1.892 *10^16 m Δt_p = 3.154 *10^8 s 3. The attempt at a solution a.) since the passengers only have 10 years worth of food, that is by definition the max proper time interval (Δt_p) for them to reach the new planet, as measured by the passengers on the spaceship. Δt_p = L/v plugging in the length contraction relation ---> Δt_p = (L_p)/(γ*v) v*Δt_p = L_p * sqrt(1-β^2) ...... after some algebra i get β = 1/sqrt[ (c*Δt_p/L_p)^2 + 1] .... plugging in the numbers i get β = 0.196 ..... hence v = 0.196*c b.) since the distance between the earth and the new planet is given to be 2 light years, I assumed that they are at rest with respect to each other. This assumption is used when the passengers measure the time interval Δt_p = L/v. Since the ship is at rest with respect to itself, and the planets are at rest with respect to each other, that means that the planet is approaching the spaceship at the same speed that earth is receding away from the spaceship, speed v. c.) L = (L_p)/γ .... plugging in the numbers L = 1.86 * 10^16 m d.) There is no proper frame of reference, each is equivalent to the other by the first postulate of special relativity. The proper time is measured by the passengers on the spaceship, and the proper length is measured by those at rest on earth with respect to those on the other planet.