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Veratule

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I was really bored one day, and I decided to make up this question for fun. It's actually pretty easy and I was able to solve it using nothing but high school algebra, but what makes the question interesting is just the conceptual aspect of the question:

A team of astrophysicists are stranded on Mars, when their life-support system goes offline, leaving the astrophysicists with only a limited supply of air. Luckily there is a space-cruiser docked on planet Earth capable of traveling at speeds near that of the speed of light. Before departing for Mars, the rescue team quickly calculates at what speed they must travel in order to reach Mars the fastest, while still having the astrophysicists in good health. If they travel too slow, then they won't be able to save the astrophysicists in time. If they travel too fast though, then the travel time for the space-cruiser will be significantly less than the amount of time that the astrophysicists must wait until the rescuers arrival.

Assume t zero is the moment the rescuers leave Earth, 299792458 m/s will be the speed of light where space is assumed to be an ideal vacuum, and Mars (which varies between 56,000,000km and 399,000,000km from Earth) just so happens to be 299792458 km away (out of convenience).

t = t prime * sqrt(1-(v^2/c^2)), where t prime is time relative to conventional space, and t is time relative to the ship.

1) What is the maximum speed (m/s) at which the Space-Cruiser must travel so that it reaches the astrophysicists in the least time?

2) How long will it take the rescuers to get to Mars (minutes)?

3) How long will the astrophysicists have to wait until the rescuers get there (minutes)?

4) How long would it take a wave traveling at c to reach Mars?

5) What is c/(velocity calculated in part a)?

I hope you guys/girls enjoy this problem at least a little bit. I know I enjoyed both writing it and solving it :) I'll message anyone who wants to know the answers (to check their work, or is just curious.)

A team of astrophysicists are stranded on Mars, when their life-support system goes offline, leaving the astrophysicists with only a limited supply of air. Luckily there is a space-cruiser docked on planet Earth capable of traveling at speeds near that of the speed of light. Before departing for Mars, the rescue team quickly calculates at what speed they must travel in order to reach Mars the fastest, while still having the astrophysicists in good health. If they travel too slow, then they won't be able to save the astrophysicists in time. If they travel too fast though, then the travel time for the space-cruiser will be significantly less than the amount of time that the astrophysicists must wait until the rescuers arrival.

Assume t zero is the moment the rescuers leave Earth, 299792458 m/s will be the speed of light where space is assumed to be an ideal vacuum, and Mars (which varies between 56,000,000km and 399,000,000km from Earth) just so happens to be 299792458 km away (out of convenience).

t = t prime * sqrt(1-(v^2/c^2)), where t prime is time relative to conventional space, and t is time relative to the ship.

1) What is the maximum speed (m/s) at which the Space-Cruiser must travel so that it reaches the astrophysicists in the least time?

2) How long will it take the rescuers to get to Mars (minutes)?

3) How long will the astrophysicists have to wait until the rescuers get there (minutes)?

4) How long would it take a wave traveling at c to reach Mars?

5) What is c/(velocity calculated in part a)?

I hope you guys/girls enjoy this problem at least a little bit. I know I enjoyed both writing it and solving it :) I'll message anyone who wants to know the answers (to check their work, or is just curious.)

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