How Can Astrophysicists Be Rescued from Mars Using Special Relativity?

[Veratule]

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.)

 

[DaveC426913]

I'm confused. What is wrong with the space cruiser simply making the trip at maximum acceleration?


i.e.:

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.

So what if the rescuers experience a trip duration of, say, 10 seconds?


Answers:

A1: As fast as the cruiser is able to go. Max positive acceleration, max negative acceleration. If it has virtually instant acceleration, then its average speed will be slightly less than c.

A2: From Earth's or Mars' FoR, a little more than 1000 seconds, depending on the answer to A1. As for rescuers, it is impossible to determine their experience without you specifying the acceleration of the ship.

A3: See A2.

A4: Exactly 1000 seconds.

A5: Slightly more than 1.

There appears to be a hidden (mistaken) assumption in your understanding of relativity.

 

[Janus]

I too am confused. The time differential between ship and Mars has no effect on how long the Astrophysists have to wait. If the ship travels at 0.866c, They will have to wait 19.25 mins for the ship to arrive. (From the crews view the trip will take half as long.)

At .99c the astrophyicists wait for 16.84 mins and for the crew it will take 2.4 min.

Oh, wait, I think I see where the confusion is. The OP has mistakingly assumed that the trip time for the crew is always equal to 299792458 km divided by their speed. Then he applies the time dilation formula which makes it seem that the trip time for the Astrophysicists increases the closer the ship travels to the speed of light.
He neglected to apply the length contraction to the distance between Earth and Mars That the crew of the ship would see. IOW, At 0.886c the distance contracts to 299792458/2 km for the crew and at 0.99c to 299792458/7 km.

 

[DaveC426913]

The upshot is that if we run simulations, each time increasing the speed closer and closer to c:

- the duration for the astrophysicists would asymptotically approach 1000 seconds, but never get less.
- the duration for the rescuers would asymptotically approach zero (but never get less ).

 

[Veratule]

Now I am equally confused. Let me just post my math, b/c it's certainly possible that I made a mistake.

According to my calculations, if you were traveling at 299792458 m/s, then you would arrive at Mars in 1000 seconds, space-cruiser time, or 16.6666 minutes, (the minimum time to reach Mars we'll say).

If we travel at 0.5c (149,896,229 m/s), then it would take the ship 2000 seconds in ship time to reach Mars from Earth (33.333 mins). And the astrophysicists would have to wait 33.333 mins * 1 / (sqrt(1-(149896229^2/299792458^2))) = 38.49 mins for the ship to arrive.

If we travel at 0.9c (269813212 m/s), then it would take the ship 18.518 mins in ship time to reach Mars, and 18.518 * 1 / (sqrt (1-(269813212^2/299792458^2))) = 42.48 mins would pass before the space-cruiser arrives to help the astrophysicists in Mars time.

The critical point is when you are traveling @ sqrt(0.5) times the speed of light (211985280 m/s), because if the ship travels any slower or faster, then the astrophysicists will have to wait longer. At this speed, it takes 23.57 mins ship time, and 33.333 mins Mars time to arrive.

Perhaps I have made a mistake. I noticed Janus said 2.4 mins travel time, so I may have this Time Dialation thing backwards, but according to my calculations, if the ship is traveling @ 0.99c, then the ship will take 16.84 minutes to get there (ship time), and 846.2 mins will pass on Mars before arrival.

Equation I used:

change in (t of the ship) * 1 / (sqrt (1 - (v^2/c^2))) = change in (t of the planet)

or

change in (t of the ship) = (sqrt (1 - (v^2/c^2))) * change in (t of the planet)

 

[DaveC426913]

Now I am equally confused. Let me just post my math, b/c it's certainly possible that I made a mistake.
change in (t of the ship) = (sqrt (1 - (v^2/c^2))) * change in (t of the planet)

No. You misunderstand the application of the formula.

Duration as measured by the Astrophycist is not altered by relativity. At .5c, the ship will take twice as long as light to reach Mars. Period.

At .5c the ship would take about 2000 seconds to reach Mars as per the Astrophysicist's measurement and about (2000/1.15=) 1740 seconds as per the rescuer's measurement.
At .9c the ship would take about 1111 seconds to reach Mars as per the Astrophysicist's measurement and about (1111/22.4 =) 485 seconds as per the rescuer's measurement.
At .999c the ship would take about 1001 seconds to reach Mars as per the Astrophysicist's measurement and about (1001/2.29=) 44 seconds as per the rescuer's measurement.
At .9999999c the ship would take about 1000 seconds to reach Mars as per the Astrophysicist's measurement and about (1000/2236=) 22 seconds as per the rescuer's measurement.

The faster they go, the closer the Astrophysicist's measurement will be to 1000 seconds, and the closer the rescuer's measurement will be to zero.

(P.S. http://www.1728.com/reltivty.htm" to calculate the dilation factor)

 

[Veratule]

Now it all makes sense.

Thank you for correcting my error. Dang, so much for all that math. Ah well, I would rather be wrong and learn from it. Thank you for correcting this mistake of mine =)

 

[DaveC426913]

SR has difficult concepts that few easily comprehend.

 

[Veratule]

I'm a Biochemistry undergrad, so I don't feel bad for not understanding it. I thought I was really smart for waking up this morning and realizing, "Woah, I wonder..." The way you described actually makes a lot more logical sense than what I was trying to describe in my own question. Glad you helped clear that up :D.