u/DotBeginning1420

Why can't you escape a Black Hole?

(Can you figure it out in 20 seconds)?

Imagine a beam of light. To get from one point to another one, it always chooses the path that would take it the least time. Or in other words, each a beam of light you see has a GPS telling it the quickest path from its current position to other point!

This is Fermat's principle and it is equivalent to claiming that light travels through materials according to Snell's law.

I tried proving Snell's law by assunimg Fermat's principle. I think it is not so hard knowing some basic geometry, trigo and calculus, but is my proof right? Something about the function I used and how to interpret it got me to some trouble. I don't see intuitively how the solution is indeed minimum.

In the diagram you see DC as the transition between two materials, DC = d, the light travels from A to B, through point O that can be anywhere on DC. In AO it travels at v1, and in OB at v2. We want to find the path that takes the least time. Each path can be expressed as OC = x (0≤x≤d). We also label the angles of the vertical lines as 𝛼1 and 𝛼2.

At the bottom you can see that we get v1sin(𝛼1) = v2sin(𝛼2), which also means that for n1=v1/c, n2=v2/c, it is the same as saying n1sin(𝛼1) = n2sin(𝛼2), both basically represent Snell's law.

However, as you can see t(x) is a linear function with respect to x, and it's hard to me to see why the solution I got should be minimal (maybe it is not?).

Our blood is made of water, which enters into our body when drinking, and being excreted out when urinating. This means that at some point all of our old water molecules in the blood might be excreted out, and being all replaced by new water molecules. How long can it take?

Assumptions:

1. The average adult human blood volume is generally the same across the days. It can be estimated by the weight height and gender. Blood Calculator

2. The average healthy adult human excretes out around 1-2.5 liters out as urine a day (depending on mainly how much water they drink).

3. The blood stays homogenous after drinking or urinating.

(One possible) approach:

>!We have v_total, v_replace, v_old_n and v_new_n. On day 1, v_old_1 = v_total, v_new_1 = 0. On day 2 v_old_2 = v_total-v_replace, v_new_1 = v_replace. Then on day n+1 we have v_new_(n+1) = (v_old_n/v_total)*v_replace. We can translate this recursive expression into an iterative one:!<

>!v_new_n = v_total - v_total*(1 - v_replace/v_total)^(n-1)!<

>!We can see that v_new approaches v_total when n goes to inifinity (but it never becomes 100% new). However we can still choose a threshold of 95% or 99% for the new volume to be essentially new.!<

>!For instance, for an adult human which have v_total = 5.5 liters, v_replace = 1.5 liters, for the blood to be 99% new: v_new_n >= 0.99*v_total=5.445.!<

>!5.445 = 5.5 - 5.5(1-1.5/5.5)^(n-1) => 0.01 = (8/11)^(n-1) => n >= 15.46.!<

>!After 16 days over 99% of the water in the blood was replaced by new molecules.!<