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 t2Solving for time (t) gives us (√ is the square root symbol)
t = √(2d / a)...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.
t = √(2 * 478 ft / 32 ft/sec2) = √29.88 = 5.4658 sec
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 = atNow, 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:
v = 32 ft/sec2 * 5.4658 sec = 174.91 ft/sec
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.