Physics of marriage and hypoxia

Abstract:

Got married, took a hike,

got high, got dehydrated and almost suffocated.

oh and a gerbil.

I lied.
I said the next post would deal with how being fat can save your life, and there are actually several ways that can happen. but that will have to wait for another instance.

A couple of years ago I got married.
Yes. I, like loads of better men than I, donned the world's smallest handcuffs, and walked the "green mile" as it were of single men, to happy cheers of "dead man marrying".

his: +5 ring vs. reason&accountability.
hers: one ring to rule them all and in the darkness bind them.

Well, actually I count myself lucky. most men get fucked on the deal, but as for myself, I got myself a ninja for a wife and a Jedi knight to boot. You should see her performing Jedi mind tricks on me or using The Voice on people - damn she's good!

Anyway after about a year of marriage we decided it's time to go for our much overdue honeymoon.
so somehow we came up with the obscene amount of money it now costs to fly to the USA, and we went to visit grandma, the family and take a hike. no really. we traveled California's national parks and hiked some...
at some point we even had oxygen shortage due to height sickness.

What had happened was, we trekked from Camp White Wolf to Ten lakes in Yosemite national park, and back, in one day.
Now Camp White Wolf is at $\approx 2,400 m$, and the mountain overlooking Ten lakes is at $\approx 3000 m$. so all in all it doesn't seem like a distinctive height difference right?
BUT (and it's a big butt for sure!) being overzealous and not so experienced hikers we spent the night near Briceburg which is at $\approx 350 m$.

Now for a bit of physics:
Humans usually breath air which is comprised among other things of roughly 21% $O_2$.
Assuming air is an ideal gas we get :
$$PV=NRT$$
Which means in a cubicle meter at sea level pressure, at $298^\circ K$, we get $\approx 40.9 {} moles$ of air which translates roughly to $8.6\text{ moles of }O_2$.

Now we will consider a thin spherical layer $\Delta r$ of gas:
$$A(r)\cdot P(r)-P(r+\Delta r) = V\cdot\rho (r) \frac{GM_{\oplus}}{r^2}$$
$$\Rightarrow -4\pi r^2 \Delta r \frac{\partial P}{\partial r}=4\pi r^2 \Delta r \rho (r) \frac{GM_{\oplus}}{r^2} \Rightarrow -\frac{\partial P}{\partial r}= \rho (r) \frac{GM_{\oplus}}{r^2}$$

For all intents and purposes the gravitational acceleration on the earth's surface and at 3 km differs by a factor of $1 \pm 10^{-4}$ so really there's no need to take into account the gravitational change in that small a difference in distance.
Thus we will treat the gravitational potential as $\Phi_N=gr$, and thus the force per kg is directed downwards and is given by $F_g=g\hat{r}$.
$$\rho = \frac{N\cdot m}{V} \Rightarrow \rho=\frac{Pm}{RT}$$
and so we quickly get:
$$\frac{dP}{P}= -\frac{gm}{RT}dr$$
Which assuming a constant temperature (which is a WRONG assumption as we will see immediately) we get:
$$P=P_0\,e^{-\frac{gmr}{RT}}$$
Where $m$ denotes the molar mass of air, $R$ the ideal gas constant, and $T$ the temperature.

Now using this formula we get that the air pressure at 3000 m above sea level is $\approx 52595$ pa. which means the amount of moles in a cubic meter in 3000 meters of height is 21.2 moles. keeping in mind that $O_2$ is only slightly heavier the $N_2$ we allow ourselves the assumption that the $N_2\sim O_2$ ratio in 3000 meters is similar to that of about ground level we get that the air up there has only ~11% oxygen in the same volume.
Just for the sake of good form - at 15-19% you get decreased ability to work strenuously.  impaired coordination may occur and there is a chance of induced early symptoms with individuals that have coronary, pulmonary, or circulatory problems. at 12-15% respiration and pulse increases; impaired coordination, perception, and judgment occurs. , at 10-12% respiration further increases in rate and depth; poor judgment and bluish lips occur. At 8-10% symptoms include mental failure, fainting, unconsciousness, an ash-colored-face, blue lips, nausea, and vomiting.

you get the picture right?

So 11% is quite close to being knee-deep in shit.

Let's have a more realistic estimate though, because we know the temperature decreases with height (at least in the atmosphere).

So basically the temperature is a physical function and we like to think of those as analytical so the first approximation of $T$ around $T_0$ is given generally by:
$$T(r)\approx T_0 - ar$$
The minus sign owing to our understanding of temperature decreasing with height.
and so the above differential equation becomes:
$$\frac{dP}{P}= -\frac{a\,gm\,dr}{a\,RT_0\left(1-\frac{ar}{T_0}\right)} \Rightarrow P=P_0\left(1-\frac{ar}{T_0}\right)^{\frac{gm}{aR}}$$
And so using the corrected formula with $a$ denoting the linear-approximate rate of temperature decline in the atmosphere which is given by $a=6.5 \cdot 10^{-3}$ we get the pressure up there was about 70970 pa which is much likelier, whereas the effective air percentage there is almost 15%.

Meaning that we we're only in an ankle-high depository.

Just as an additional reference, in normal circumstances the oxygen percentage in our exhaled breath is about 16%, and in some circumstances could go as low as 14%.
So imagine yourselves in a hot, non-ventilated room, stuffed with cheap foreign workforce, and imagine the wicked headache you'll get. Now multiply it, as we were climbing up at a quick pace thus taking even shallower and quicker breaths.

At some point we actually sat there at the top of the mountain, breaking for lunch, not realizing we are only making matters worse...

The payoff was worth it though - it's such a beautiful view up there - it's AMAZING!

Ten Lakes view, headache and dehydration included.

Anyway, on our way back we also came to the undeniable conclusion we were idiots, as we ran out of water, we had a pounding headache, and we had to virtually run all the way back to Camp White Wolf before dark, since we were really afraid to walk in the woods at night, with no light, no fire, no reception, and loads of wolves howling in the distance.

We got there in the nick of time,  the sun was already setting as we hit the marked trail back to camp, about 20 minutes later, in full dark, we got there.
Sore-foot, dead-tired and otherwise completely wretched, we decided that we are NOT making camp to sleep on cold bare ground. I drove us to a nearby motel outside the national park, and we took a shower, and slept the deepest sleep achievable by mankind...


oh, and a gerbil:

Gerbil

Next time: Lenses and idiots.

Physics of falling from a second floor window

Abstract:

Fell from second floor, 

and the first thing I thought of was physics!

oh - I also crashed into a parking semi-trailer.

A long long time ago and in a galaxy not so far away, though it does seem like a whole different life to me now, I lived in Jerusalem, in the then new student dorms situated in mt. Scopus.

One fateful Sunday, after returning  from a weekend at my parents house, I found myself locked out of my second-floor apartment. a quick search through my pockets yielded little but lint, and pretty soon the reality of forgetting my apartment keys in my parents house dawned on me.

Now ordinary people might contact the maintenance person, get a spare key, or - I don't know, call their roommates or something creative like that.

But it seems that either prolonged army service fucks your brains up, or maybe I was dropped on my head one time too many when I was a kid...

Instead of the logical path, I took the "highway" literally. I climbed up to my second floor using nothing but my hands. I got up on a nearby trashcan, and jumped to hang on a ledge by my bear hands, proceeding to pull myself up to our outside porch. from there it was a pretty easy business to climb the rest of the way using the stonework to the second floor porch window, opening it (it was a sliding window and those are REALLY easy to open if you know the trick), and getting in my warm and fuzzy apartment.

Now keep in mind, I am a heavy guy so this is no simple feat.

Anyway, the next day, as if it was even possible, I was even dumber!!! instead of asking one of my roommates for an extra key, I slept in with the immediate result of getting myself locked in...

Being a smart bugger, I tried to climb down the way I clambered up... and to my surprise, climbing down proved to be A LOT EASIER than climbing up... I just fell down.

Now again to illustrate two points:
1. I'm heavy and quite big - this will play a significant role later in this post.
2. The mind works in mysterious ways, specifically, my mind is a little bit warped.

Now the first point is probably responsible for the fact that I am alive today, more on this later.
The second point is directly responsible for this post! as the following physics consideration and their astonishing result is the FIRST thing that went through my mind as I was hitting the ground.

So lets start shall we?

some data first:

• the standard floor-to-floor height in western countries is about 3 meters.
• my weight at the time was about 100 kg.

and so from Energy conservation consideration:
$$E_{pot}=E_{kin}\Rightarrow gh=\frac{1}{2}v^{2}\Rightarrow v_{impact}\approx\sqrt{120}\,_{\frac{m}{sec}}\approx 40\,_{\frac{km}{hr}}$$
So basically falling from the second floor means you hit the ground with a velocity of about 40 Kph. And the momentum at which I arrived to that fateful event is given by:
$$mv_{impact}=100 kg \times \sqrt{120}\, \frac{m}{sec}\approx 1100 \, \frac{kg\cdot m}{sec}$$
And by Newton's second law:
$$F=\dot{P}=\frac{\partial P}{\partial t}\approx \frac{P}{\Delta t}=2200\,N$$
with $\Delta t$ set to say half a second right? we all remember slow-mo pictures of people get hit in the face - that's about a 1000 frames per second and so half a second sounds about right...
for the sake of proper reference an m-16 bullet has a barrel velocity of about 905 $\frac{m}{sec}$ where the bullet's mass is about 4.1 grams, and we end up with a measly 37.105 $\frac{kg\cdot m}{sec}$, and given the event of a hitting bullet takes about 0.01 seconds we get a force of about $3700\, N$, which is comparable to the previous result.

So now one would ask why the hell didn't I die right there and then?
well the answer has actually two parts:

1. it's not just the force that counts. it's the pressure that will determine if my skin raptures or not and a bullet has a very low area of interaction with a body, thus the pressure at the tip is immense, and the bullet is able to enter the body, where the body's consistency versus the bullet's energy determines if and when the bullets go through, or stays in (for the night) and discharges all it's energy thus making a lot of damage. - hmm this actually gives me an idea for a future post on bulletproof vests...
2. I fell straight on my back thus enlarging my area of interaction. had I fallen head first, or in such a way as to present a thinner profile, or simply falling in a position to hurt essential organs, I would not be writing this post now...

Yet another nice  comparison to make is that of a car-crash:
Consider a car going at about 100 kph, on a wet highway, near Keisarya.
Suddenly the driver in front of our car brakes, and our car brakes and veers right towards the shoulder, only to find there's a parked semi-trailer there.
Our car then proceeds to crash head-on with the truck's aft, at about 80 kph (remember, the car braked, so it slowed a little).
Some of the impact is absorbed by the car's front end, and the whole event takes about 1 second.
thus, the driver's body, that weighed around 100 kg just then has a momentum of about
$$P=100kg \times \frac{80}{3.6}\approx 2200 \frac{kg\cdot m}{sec}$$
And since the event takes about a second the total force is given by $F\approx 2200 N$

Notice the similarity in both cases, where the force on my body was comparable and the total result almost the same:
in the first case it took my body about a day to recover from the first adrenalin rush, it took me to that day's end to feel the pain, and then about 2-3 days of recovery after which I ached for about a month.
in the second case, the first day was spent in emergency ward checking that everything is in it's place and the pain came the morning after, incarcerating me in bed for 3-4 days with a month's worth of inability to practice due to - you guessed right! - after effect aches...

next time: how being fat can save your life!