μ-mesons (muons) rapidly decay, their mean lifetime is 2.2 μs.
In our case μ-mesons are produced at the reactor and must be delivered via a vacuumed pipe to the laboratory. The reactor and the laboratory are separated by distance 1000 meter and are both located at see level, and μ-mesons travel at constant speed c/2 through the pipe. We can bend the pipe in any shape, and deflect the muons with magnetic field, if necessary.
We want to deliver as many whole un-decayed muons as possible, which means of course minimizing proper time of travel.
Shortest distance would be straight line, connecting the reactor and the lab, but better results could be achieved if muons follow curved trajectory slightly above the straight line, because due to gravity g time runs faster there.
What is maximum elevation of the optimal trajectory above the straight line?
Answers & Comments
Verified answer
Does see level mean you can see them? Reminds me of the old amtrack slogan--see America at see level :)
As for your question, I suspect the quickest path would be a freefall trajectory. If you don't wish to actually calculate that parabolic path, the smartass answer would be to invoke the notion that gravity is warping space, but that the freefall path is truly a straight line.
It may help to imagine that instead of being in earth's gravity g, you are in a rocket accelerating through inertial space at a rate g. You send the muon a distance d at a speed 1/2 in a time of (to first order) t=d/2. During that time, the rocket hurtles forward a (tiny) distance of 1/2 gt^2. So you want to shoot a straight line (in flat space), whose length is (by pythagorus):
length = sqrt (d^2 + (1/2 gt^2)^2).
And of course in the rocket (or on earth) the trajectory is that straight line superimposed on the constant acceleration 1/2 gt^2.