You are exploring a newly discovered planet when a colleague contracts a deadly alien virus. Left untreated, it will probably kill him in 5 days, but the nearest medical station is 10 light-days away! How can you get him there in time?
\gamma = 1/sqrt(1-v^2/c^2)
The Attempt at a Solution
Intuitively, I read this question and concluded that you could not do it. How could you possibly go 10 light-days in only 5 days? However, I tried to work through it anyway. Given the equation for Δtp (proper time), I substituted some other values. The L term represents the contracted length, so I replaced L with L'/\gamma into the equation Δtp=L/v. This gave me Δtp=[(L'(1-v^2/c^2)^.5)]/v. Doing quite a bit of algebra, I ended up solving for v (the velocity of the ship) and got v=(L'^2/Δtp^2)^.5.
I converted 5 days in to seconds (so that I could work in meters/second) and got it to be 432,000 seconds. I said that this should be my Δtp term because this is the time measured on the ship, so you need it to be 5 days to save the other astronaut. I then converted 10 light days in to meters (again, to keep things simpler for me). I found 10 light days to be 2.592E14m. Plugging this in to my previously derived equation, I got the speed necessary for the ship to reach the medical station (including time dilation and length contraction) to be 2c.
Is this supposed to happen, or should there be a true answer that would allow me to get the other passenger to travel 10 light days in a 5 day span (in his frame of reference)?
Any help would be much appreciated.