## Wednesday, May 09, 2007

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### Falling, yes I am falling...

Forgive the somewhat macabre source of this entry (and forgive the link to the Daily News, but the AP story that ran in the Times is much more mundane). About a month ago, a man jumped from the Empire State Building in New York City:

A lawyer leaped to his death from a 69th-floor office at the Empire State Building yesterday [...]

[...] Police said the rest of Kanovsky's body was found intact on a 30th-floor landing.

When it happened, I had a discussion with a friend about the speeds and times involved. It came up again the other day, in a discussion with someone else, and I thought I might write it up here. The numbers can be rather surprising.

The first question was how far he fell and how long that took. “It must be terrifying,” my first friend had said. “Imagine that you've jumped, and then you have a second thought about it and wish you could take it back. Now you have to think about that all the way down!”

So let's see how long he had to think about it, using approximate, back-of-the-envelope calculations. First, the distance: the Empire State Building has 102 floors, and is 1250 feet high at the 102nd floor. Mr Kanovsky fell from floor 69 to floor 30, as the excerpts above say, a fall of 39 floors. Let's assume that the floors are all of equal height and spacing (surely not true, but close enough):

d = (1250 ft / 102 floors) * (69 - 30) floors = 477.94 ft
...so he fell about 478 feet.

A falling object near the surface of the Earth has a constant acceleration due to gravity — its speed increases by about 32 ft/sec each second (a = 32 ft/sec2). For a relatively small object with a relatively short fall, the effect of air resistance is negligible, so let's omit that from our calculations. The formula for the distance travelled from a starting velocity of zero and with a constant acceleration is

d = ½ a t2
Solving for time (t) gives us (√ is the square root symbol)
t = √(2d / a)

t = √(2 * 478 ft / 32 ft/sec2) = √29.88 = 5.4658 sec

...so it took him about five and a half seconds to fall. Five and a half seconds to think about it. Is that a lot, or a little, when the ground is moving toward you at... well, at 32 feet per second faster every second? I don't know.

But that brings up the next question: How fast was he moving when he hit the 30th-floor ledge? That's the simplest formula of all. Since the acceleration is constant:

v = at

v = 32 ft/sec2 * 5.4658 sec = 174.91 ft/sec

Now, 175 ft/sec doesn't mean much to most of us, so let's convert it to miles per hour, which we Americans can better get our heads around:
174.91 ft/sec * 3600 sec/hr / 5280 ft/mile = 119.25 mile/hr
...and so the poor man reached the ledge at the surprising speed of about 120 MPH.

My guess is that he didn't know what hit him.

[Just to complete the picture, someone jumping out of the top floor and hitting the ground would traverse the entire 1250 feet in 8.8 seconds, and would reach the ground at a velocity of about 193 MPH.]

Update, 9:35: See Paul's comment in the comments section. Summary: even for calculations on the back of the envelope, it's not OK to ignore the air resistance for a falling human.

Paul said...

Actually, the effecet of air resistance cannot be removed from the equation, as it is significant on a human body, especially one that is fully clothed, and not making any special effort to reduce one's profile. Terminal velocity of about 120mph is reached in about 8 seconds of free fall, so our jumper probably did not quite reach the speed you mentioned, but he probably came pretty close. If he removed his jacket before jumping, and intentionally held his arms and legs closely together, his terminal velocity might have increased to about 200mph, and he likely would have reached that 119mph before he hit the thirtieth floor, but I suspect that is an uncommon occurrence for a jumper.

Dr. Momentum said...

Maybe I have a macabre interest, but I was quite keen to find out how many seconds he had to think about his decision.

I expect the wind racing by was somewhat of a distraction.

The post reminded me, of course, of the old joke about the guy falling form the top of a skyscraper who is asked "how is it going" and replies "so far, so good."

There are many possible analogies. I leave them up to the reader.

Barry Leiba said...

And James just made me think about the bit from... one of the Hitchhiker's Guide to the Galaxy books, I forget which... in which the dolphin is plummeting to the ground, and hearing the wind whoosh by.

Miss Profe said...

Certainly a different perspective on a jump from a building.

Of course, the religious question always is: Did the person repent as he/she was falling? Did he ask forgiveness as he/she fell?

Barry Leiba said...

Repent for what? For jumping? Hm, that's interesting: can one atone for making an irreversible decision?

Yoshiko said...

>the dolphin is plummeting to the ground, and hearing the wind whoosh by.

I think that's the first one. I am not so sure though. ( I love the books, but unfortunatelly, only three books are translated into Japanese. Maybe I will try to original ones. )

Barry Leiba said...

I imagine that the Hitchhiker books would be tough to translate without losing some of the humour... and I similarly imagine that they'd be tough to read in English if you have trouble understanding funny turns of phrase. But it's certainly a good thing to try! I think the first three books, in particular, are wonderfully funny. The others (So Long, and Thanks for All the Fish and Mostly Harmless), as well as the "Dirk Gently" books, are less good, in my opinion... but still worth reading.

Paul said...

I believe, Barry, it was a Sperm Whale. And a flower pot. Although I'm just going by memory, here.

Barry Leiba said...

Yes, thank you, Paul! Your memory's better than mine. Exactly, a sperm whale and a flower pot. (I'd moved the dolphins over from another part of the story.)

Miss Profe said...

Repent for jumping/commiting suicide.

Can one atone for making an irreverible decision? Yes: asking for forgiveness.

scouter573 said...

I'm not sure he lasted the full 5-8 seconds. The fall was more traumatic than seems at first glance. Note that the report says something about "the rest of Kanovsky's body..." Makes one wonder what happened to the other bits. May he rest in peace.

Barry Leiba said...

Oh, as I interpret it, his body hit the 30th-floor ledge, at which point his leg, which was perhaps sticking out from the ledge, was severed and continued to the ground. My interpretation is that his body was intact until it hit the ledge.

It's possible that that's not correct. In any case, I think I don't want to know any more detail — I'm not into "gruesome".

Ray said...

Please forgive the poor taste, but I simply couldn't let this pass...

The fall was more traumatic than seems at first glance. As in the glancing blow caused by his meeting the ledge?

May he rest in peace. Uh, shouldn't that be pieces?

Well, I did apologise...

An Qi said...

I tried to recall my “free fall” rides at Six Flags several years ago. According to my experience, you don’t really think about anything when you fall like that. The physical discomfort is so intense that your mind just goes blank.

But I wonder if he “aimed” for the streets when he jumped. I’d hate to see innocent people get killed that way.

Barry Leiba said...

I think that the possibility of injuring people on the ground is something that jumpers seldom think about beforehand, but it's a real possibility. And that possibility has shut down the Tappan Zee Bridge, a major commuter route north of NYC, at least twice in the last few years. During the ironically named period of "rush hour", on two occasions people climbed the superstructure of the bridge and made as if they'd jump. Traffic was cleared from the vicinity of the jumper, lest he land on a car and injure the driver.

The "free fall" rides are actually not usually free falls. Because of how they work, you have downward acceleration imparted at the start, so you're accelerating at more than 32 ft/sec/sec and experiencing more than the normal 1g force. Which is why it feels so icky. :-)