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!