How Far Do You Fall In One Second? Two Seconds?

[DonRocks]

"Falling" (1967) by James Dickey (1923-1997, author of "Deliverance")

At the end of this post, you'll be able to answer these questions nearly instantly, and you'll remember how to do it for the rest of your life. What I'm about to tell you is no more advanced than what a middle-school child learns in science class (and forgets the moment the test is over).

We all know the names Pythagoras and Galileo.

Pythagoras (c570BC - c495BC) is most famous for the Pythagorean Theorem (which has nothing to do with this). He was a Greek scholar, philosopher, and mathematician, and was clever enough to devise the following mathematical formula. Don't stop here - this is *easy*!

Note that the drawing is a "square."

The numbers on top are just 1, 2, 3 ....

The numbers on the left are the number of dots in that section (count them and see).

You should be able to clearly see that this drawing can be extended to infinity. But what does it represent?

Let's take the number 3. Count up all the dots in sections 1-3, and you'll get 9 dots, or, 3-squared.

With the number 4, count up all the dots in sections 1-4, and you'll get 16 dots, or 4-squared.

This is all very easy to see, and intuitive as a graph; unfortunately, it needs to be represented as a formula. Don't leave! Skip the following line if you need to because it's not that important:

For any number (call it "X"), it's square is equal to the first "X" odd numbers, added up. Don't leave!

With the number 4, it's square is equal to the sum of the first 4 odd numbers: 1 + 3 + 5 + 7 = 16.

Hi-Fi was rumored to be a square as well:

[Exit Pythagoras]

[Enter Galileo]

Galileo (1564-1642) is one of absolute most famous scientists in history, and his accomplishments are so vast that listing them here would be pointless. There really isn't any "one thing" he's most famous for; he's a lot like Leonardo da Vinci - just a total Renaissance man, and you'd have to put him on any Top 10 list of "Scientific Contributions To Mankind" for his lifetime achievements.

Galileo was fascinated by Pythagoras, and one of the things he did was take this formula by Pythagoras - purely mathematical - and apply it to the real world. In other words, he took pure Math, and applied it to Physics.

Galileo figured out that the above figure corresponded almost exactly to how fast objects fell. This is what he figured out. Don't leave! This is just as easy.

Here are a few details that you can skip because for the purposes of understanding this, you don't need to know them; just be aware that they exist:

SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME 

1. In reality, this applies only to objects falling in a vacuum. Things like drag (stick your arm out the window of a moving car) and buoyancy (a cork floating on water) are important to scientists, but not for us.

2. All things - no matter what their weight, mass, or density - fall with the same acceleration and speeds. This has been proven, and you can count on it being true: in a vacuum, a feather will fall exactly as fast as a brick, and they'll hit bottom at the exact same time.

3. There is an upper-bound called terminal velocity which happens when the forces of drag + buoyancy cancel out the force of gravity. Since you've made it this far, you are hereby rewarded by the trailer of the 1994 film with the same name:

SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME 

Are you still here? Okay, we're almost done. Don't leave!

Here is what Galileo figured out, using what Pythagoras did as a basis (note that the metric system was not introduced until 1668, after Galileo's death):

For every 1/4-second increment spent falling, you cover the distance shown by adding up the numbers on the left side of the above figure. Examples:

1/4-second: You fall 1 foot.

1/2 second: You fall 1 + 3 feet.

3/4 second: You fall 1 + 3  + 5 feet.

So for every 1/4-second interval that something falls, just add up the odd numbers. That's it!

Now, ask yourselves: how far do you fall in one second? Two seconds?

Hint: one second is four 1/4-second intervals; two seconds is eight 1/4-second intervals.

(The answers are 16 feet (1+3+5+7) and 64 feet (1+3+5+7+9+11+13+15), respectively.)

As a shortcut which makes it even easier, you can just take the square of the number of 1/4-second intervals (for one second, it's 4-squared; for two seconds, it's 8-squared; for 5 seconds, it's 20-squared which is greater than the length of a football field).

You are now free to live the rest of your life knowing that if you fall for much longer than one second, you're pretty much fucked.

PS - the sheriff at the end of Deliverance was James Dickey himself:

Does anyone know why Dickey gets in and drives off in the passenger's side of the car? Is this some weird mirror-image thing? Or was this filmed in England?

[Bart]

2. All things - no matter what their weight, mass, or density - fall with the same acceleration and speeds.

What about soaring birds that go up, down or stay at the same altitude without flapping their wing?

They're falling too, it's just the air around them is rising faster then they are falling (plus the wing and feathers help), but they don't float!

[DonRocks]

What about soaring birds that go up, down or stay at the same altitude without flapping their wing?

They're falling too, it's just the air around them is rising faster then they are falling (plus the wing and feathers help), but they don't float!

You seem like you understand this, so I assume your first question is rhetorical. But, just so everyone is on the same page ... lift, which is a variation of drag - in a vacuum, there's no drag, no lift, and no soaring birds.

Lift is something I've never completely understood, but it's "in contrast to" drag. I think lift is why, when you hold your arm out a moving car window, you can make your hand "soar" towards the wind by turning your palm down to the ground.

I remember when Matt was about 12, I was explaining relativity to him (in the most basic of ways). We were driving on the outer loop of the Beltway near the Wilson Bridge (I don't know why I remember that), and I told him the tennis ball in his hand was traveling at 60 mph (if you take into account the earth's rotation, revolution, galaxies revolving around each other, etc., it's moving one heck of a lot faster than that). Well anyway, as an experiment, he opened the sunroof, and tossed the ball straight up: it instantly disappeared, and, feeling guilty, I made sure we each picked up five pieces of trash that day. As for the driver behind us, I did not ask him his opinion (I knew of a driver who claimed to keep a few used spark plugs (!) in his car to fend off aggressive trucks (porcupine, did you ever watch "Duel?") - I also know of a motorcyclist who claimed to *shoot* at the back of a pickup truck who cut him off - I don't go there, which is one reason I'm still alive).

[Bart]

Don, I'm not sure I really understand it, but yes, that was a rhetorical question!

I was just thinking back to a class on took on Hawk ID (identifying hawks in flight) and the instructor mentioned that birds don't float.  They are constantly falling.  This sort of blew my mind at the time.  I never really thought about it that way, and it's obvious when and if you think about it............birds (like all matter) are affected by gravity so of course they don't float, but I bet you'd get a lot of weird or wrong answers if you asked the average man on the street if birds can float on air.

On the other hand, they do "float" but that has to do with wind, and thermals, and wings, and feathers.

[JPW]

Lift is something I've never completely understood, but it's "in contrast to" drag. I think lift is why, when you hold your arm out a moving car window, you can make your hand "soar" towards the wind by turning your palm down to the ground.

I always understood it as the result of differential air pressures, but then again, I am not an engineer.

I did however stay in a Holiday Inn a few months ago.

[DonRocks]

Don, I'm not sure I really understand it, but yes, that was a rhetorical question!

I was just thinking back to a class on took on Hawk ID (identifying hawks in flight) and the instructor mentioned that birds don't float.  They are constantly falling.  This sort of blew my mind at the time.  I never really thought about it that way, and it's obvious when and if you think about it............birds (like all matter) are affected by gravity so of course they don't float, but I bet you'd get a lot of weird or wrong answers if you asked the average man on the street if birds can float on air.

On the other hand, they do "float" but that has to do with wind, and thermals, and wings, and feathers.

I don't think they are constantly falling - that would be like saying a rocket ship taking off is falling. What I think would be correct to say is that there's a constant gravitational pull on them, and for them to go upward, there must be a counter-force greater than the gravitational pull.

Of course, this all depends on your definition of "falling" - my definition would be "plummeting towards the ground," and they're clearly not; your teacher must have been using some other definition.

[Bart]

Yeah, you're (sort of) right.

The rocket has an engine pushing it up.  The soaring bird doesn't.  The bird's "engine" is the warm air rising faster than the bird is falling.  The point he was making was about thermals vs falling birds.  This was in the context of explaining why soaring birds like hawks don't fly at night and don't (for the most part) fly over large bodies of water.......no thermals over either one, so no rising air, so no soaring hawks.

[DonRocks]

The rocket has an engine pushing it up.  The soaring bird doesn't.

No shit. My point was only to introduce a counter-force that was easy to visualize; not to stay within the realm of fluid dynamics (since the bird is already a solid, I figured we'd already left that realm) or even aerodynamics.

SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME - SKIP ME 

I said to SKIP ME!  

Quick! How many feet does a cling peach fall in 1.5 seconds in a vacuum?

[DonRocks]

I always understood it as the result of differential air pressures, but then again, I am not an engineer.

I did however stay in a Holiday Inn a few months ago.

You know what? I just re-read this post, and this is one of the most psychedelic, surreal things I've ever read.