A precision marksman is attempting to shoot a target across a desert valley where strong thermal gradients create a mirage effect. The air temperature varies linearly with height according to T(h)=T0+αh, where T0=320K is the ground temperature, α=−0.008K/m, and h is height above ground. The refractive index of air depends on temperature as n(T)=1+β/T where β=7.8×10−5K.
The marksman's rifle is positioned h1=2.0m above ground, and the target is at the same height h2=2.0m at a horizontal distance d=800m away. Due to the varying refractive index, light rays follow curved paths in this medium.
(a) Find the equation of motion for a light ray in this stratified medium, expressing dx2d2h in terms of h.
(b) For the ray that connects the rifle to the target, determine the maximum height hmax reached by the light ray during its curved trajectory.
(c) Calculate the total optical path length of this ray from rifle to target.
Note: Use the paraxial approximation where (dxdh)2≪1.