The crystal vault.
(some lost in space back story here, will get chatbot to generate)
(data is newly generated at each refresh.)
Your spaceship lands on the planet SNELL, and you come across an ancient alien vault that contains
some knowledge artefacts that you seek. You realize to open the vault you needs use the vault laser
and shoot a red laser beam to the blue receiver through a stack of crystals that are layered on top
of each other.
Each layer of crystal has a thickness of exactly 1 unit, and there is no gap between layers.
The targeting laser beam has an initial incidence angle fixed at 45 degrees.
This is the angle the red laser beam makes with the surface of the first layer of crystals.
Each layer of crystal is made of a different material, so the laser beam has different speeds in them.
Let us denote v_0 to be speed of the laser before entering the crystals, and let v_i be the speed of the
laser in the i-th crystal layer.
A table of the ratios v_i/v_0 for each layer is given below, in the order that goes v_1/v_0, v_2/v_0, etc.
Find how many horizontal units has the laser beam been deflected from the vertical incidence line (shown dashed)
as it reaches the bottom of the last crystal layer.
Enter your answer in units here, round to four decimal places.
(part 1)
* * * part 2 * * *
To activate the blue receiver and open the vault, we need to have a horizontal deflection of 25 units from
the vertical incidence line. Unfortunately the vault laser is mounted and you cannot change its position nor angle.
Luckily you have an alien photonic supercharger that can change the speed of light in a layer of crystal.
You decide to change the relative speed v_1/v_0 of the laser beam in the first layer of crystal.
What value should you replace the first entry v_1/v_0 with, so that the horizontal deflection from the vertical
incidence line as it reaches the bottom of the last crystal layer is 25 units?
Enter your answer here, round to four decimal places.
(part 2)