Dropping Pennies from the Empire State Building

You probably heard about the idea of dropping a penny from the Empire State Building and its

probability of doing some damage. The word from various sources is - No, it won't do much of any damage. If it hit you, you might think, "Hey what was that?"

They say (they being those in charge of such things) that a big part of the reason is the flat shape of the penny and the air resistance it would meet. Granted, but what if there is no resistance? Let's try it out.

(Final velocity)² = (Initial velocity)² + 2gd is a formula for a free falling object. (Time out - I, for the first time, used html to show exponents. Wow!! Big day for me.)

Mass of penny = 3.1 grams. (I'm going with that, but I saw that it is now 2.5 grams. And I didn't even notice.) Height of Empire State Building is 86.42 meters, and initial velocity of zero. That gives us a final velocity is 86.42 meters per second or 193.32 miles per hour. But as they say, that is a moot point as air resistance is not going to allow it to go that fast. Even if it is going fairly fast, it doesn't weigh very much.

Then I thought, what if you drop a baseball and what would it's impact be? And what would be a measure of its impact. Would we be looking at momentum or kinetic energy? If I had gotten an A instead of a gift B in physics, I might have an answer for that.

Momentum = mass x velocity
Kinetic Energy = mass x (velocity)²

So take the penny's momentum/energy. What would the velocity of the baseball be to have the same impact as the penny?

To make a long story short, I figured if having the same momentum, the baseball would be going 1.488 m/s or 3.329 miles per hour.

If looking at kinetic energy, the baseball's velocity would be 28.275 miles per hour.

Neither speed would hurt a whole bunch. Hopefully no one drops something like an anvil.

10 Cool Things about Rene Descartes

This is a blog about high school applications of mathematics. However, sometimes I've thrown in some historical things or bios. I was thinking maybe once a month I might focus on a particular mathematician. It might be interesting when presenting, say the topic of analytic geometry, some info on the so-called "Father of analytic geometry" Rene Descartes. We'll see how it goe.

I thought, also, instead of a dry bio, make it into a top ten list. These won't necessarily be the 10 most vital things about the mathematician. Just some things I found interesting.

Without further ado: Ten Cool Things about Rene Descartes

1. His hometown, LaHaye, France is now known as Descartes, France in his honor.
2. He earned a law degree from the University of Poitiers.
3. he did not publicize his theories on the solar system being heliocentric after seeing the treatment Galilleo received.
4. In spite of this, Pope Alexander still put his works on a list of prohibited books.
5. He never married, although had a daughter who died at the age of five.
6. Feeling there were to many things being accepted as true, he started with only "Cogito, ergo sum" - "I think, therefore I am.
7. At the turn of the millennium, the A&E Network conducted a survey of historians on the 100 most important people of the millennium. Descartes came in #32.
8. The concept of exponents predated his time, but Descartes had the idea of writing them as superscripts.
9. He showed that a viewers angle from the center to the edge of a rainbow is 42 degrees.
10. He wrote proofs verifying the existence of God.
11. 

World Population Growth

Here is an interesting graph. (https://ourworldindata.org/world-population-growth/) It shows the world's population growth up to the present day and then someone's estimates as to what will happen in the next few decades. It probably takes a little looking at for it to make sense. I combines two graphs in one. The horizontal axis shows the passage time in years and the vertical shows growth rates. The graph also shows the total population although these numbers are just recorded on the graph rather than being recorded on the vertical axis. As line graphs go, it's a pretty busy graph.

It seems to me that math teachers could make use of the graph in pretty much any high school mathematics class.

This actually started for me with information I found in the 2017 World Almanac. It showed population estimates going much farther back in time than this graph shows. You could look at that information as a set of ordered pairs with years being represented as x-values and world population (in billions) as y-values. The almanac states that in the year one the population was an estimated 300 million. That gave me an ordered pair of (1,0.3). Proceeding in this manner gave me ordered pairs of (1,0.3), (1250,0.4), (1500,0.5), (1804,1), (1927,2), (1960, 3), (1974,4), (1987,5), (1999,6), (2011,7).

Just using the raw data, an Algebra I class might simply write ordered pairs, or without seeing the above graph, choosing appropriately labeled axes to graph the data.

Higher math classes could look at finding an equation to model the data. I had more trouble than I thought I would. I guess that is because, as the graph shows, the rate has varied over time just in the last couple centuries, let alone millennia. Leaving out the first few ordered pairs and adjusting the data such as changing (1804,1) to (0,1) and so on, I was able to find an equation that had a correlation of r = .9647. Students could maybe experiment with similar things to get a best fitting curve.

Calculus students would be able to examine the blue population growth curve and discuss how it ties into first and second derivatives. It is interesting that the person making the future projections seems to think our current point in time seems to correspond to an inflection point. Students could discuss what that really means and mathematically and socially.

Weight of a Car

I was told this weekend that you can find the weight of a car by multiplying the area of contact the tires make with the ground and the air pressure in the tires. Color me skeptical. I had never heard such a thing.

I was a doubter. I was told the area would be measures in square inches and the pressure in the typical pounds per square inch. Then I thought, area x pressure would then be sq. inches x pounds per square would give you and answer in pounds. So the labels work out, but I was still far from being convinced.

I thought I had a counter proof. What if you put more weight in the car? Then contact area would be greater and I would think the pressure would go up as well. So the car calculations would now be greater. Then I realized I started by making the car heavier, so of course the answer should be greater. So, in trying to disprove this idea, I actually acquire more evidence for it.

I looked online and found that a formula of F = PA or Force = Pressure x Area. (I probably should have remembered that from when I took physics, but that was a while ago.) Since gravity is a force, I'm becoming a little more convinced.

I looked it up online. I literally Googled "tire pressure to find weight of car" and found it. That Google is pretty smart. Anyway, they had a lesson for it. I didn't read it very thoroughly, but it seemed to back up what I had been told. It also said that when done you could check for the actual weight by looking at the sticker inside the driver's door. Well, there is another thing I learned today. I didn't even know there was a sticker there.

It seemed they found the area of a tire and multiplied it by the psi for each individual tire as there could be slight differences from tire to tire. Then they added them altogether.

How does it work out? I don't know. I might try it sometime. Right now the average high temperature is about 35 degrees. If I do spend much time outside, its going to have to be for a better reason than pursuing this. Maybe in the Spring.

How would it work though if I did do it? I know how to find the air pressure. But to find the area? Maybe outline it with chalk? What shape would that be? I'm thinking maybe like a running track with two straight segments and two semicircles. But I'm not sure.

It does seem like an interesting math application and one a class would get into. As a practical matter, probably not. If you really wanted the weight of a care, it seems easier to just check that sticker inside the door.

More Great Videos

Last time I wrote about the group OK Go. It was about one specific video. Well, they actually have a number of amazingly great videos. One you might check out is this one -

Fun, huh?

Following is one that talks about the making of the video. Its doesn't get real detailed on the math, but does allude to the ratios and proportions that must be used to come up with the final product. Granted it might not be an earthshaking important application, but students would get into it and a teacher could make use of it if study ratios at the time. For example ... "The fast portion took 4.2 seconds and the entire video is 4 minutes 12 seconds. If the exploding paint sequence lasted 11 seconds in the long video, then how long..." You get the idea.

There are a lot of hidden gems. It wasn't until I had seen it a few times that I noticed them even throwing in the Doppler effect.

It seems like there is trick photography involved, but there isn't. It takes them weeks to do these videos. Pretty amazing and it shows students the hard work they put in physically and mentally. The videos are cool and addictive - both the videos themselves and their "making of" videos.

Best Music Video Ever

Best video ever. Check it out at okgo.net. It features my now favorite band, OK Go. It's not their only

awesome video either. Did I tell you to check it out? Do so.

The band was approached by a Russian airline about using a plane to do some flights for a video that would involve weightlessness. I know. A little hard to believe. But I read about it in an issue of Smithsonian Magazine. If they aren't a credible source, who would be?

The weightless part of the flight doesn't last long and to do anything substantial, it probably takes several runs at it. You might be familiar with the concept. you can fly a plane or rocket in a parabolic path to achieve weightlessness. It's how the Tom Hanks film, Apollo 13, had its weightless moments shot.

The article (December, 2016, pages 52,53) explained the process.

1. Steady horizontal flight.
2. Hypergravity (1.5 to 1.8 g's) for 20 to 25 seconds at a 47 degree ascent.
3. Microgravity (0 g's) for 20 to 25 seconds.
4. Hypergravity (1.5 to 1.8 g's) for 20 to 25 seconds at a 47 degree descent.
5. Steady horizontal flight.

The weightlessness takes place during that 20 to 25 second sweet spot at the top of the curve. As the authour Jeff MacGregor stated "Then came the math. The song is 3 minutes and 20 seconds long, give or take. Weightlessness during parabolic flight occurs in roughly 25-second increments. That's at the top of each parabola. And for every parabola, it takes five minutes of flight to reset for the next one. To get a single continuous weightless take lasting 3:20 would require eight parabolas - more than 45 minutes of actual flying time. 

This is perhaps not the best ever math application. However, have used it in movies and videos and it has even been used for weddings. Maybe it is the best ever math application.

Book

My book is coming along. My blog is to promote the same idea as my book - applications of mathematics at the high school level. I don't want to be some jerk and use this blog to publicize my book. At least not very often. Let me take this week, though, and tell you where I'm at on it.

I just heard from my peeps at McFarland Publishing. They are pretty much done with what they do. The sent me a PDF of my book - exactly 200 pages. I now have two more jobs.

One is to do a final read through for errors. I'm about 3/4 the way through that. It seems pretty good. The main mistake that has come up is where I use a subscript or superscript in an equation and the next symbol mistakenly gets put into that form as well. No big deal. I think that is easily correctable.

They say I can respond by e-mail if there aren't too many errors (under 30) or run off the offending pages with corrections written in. They let me know it isn't time for general editing. This is just to find errors. I'm not sure I always know the difference, but in most cases I think I do.

Job 2 - Create an index. I've never done that before. I thought I would read it for mistakes and plan out the index at the same time. Nope. That turned out to be way more than my head could handle. So by tomorrow I think I'll be through the initial reading. I don't think I'll have to read every word to do the indexing, so perhaps that will go faster.

I've wondered before how they come up with an index. They sent instructions on how it is done, or at least how they want it done. It's pretty interesting. There are some rules, but at the same time a lot is left to the discretion of the author. Rules just on the correct alphabetize an index is a lot more complex than one would think.

I have to do this and send it in by December 31. It goes to the printer then. I'm not exactly sure when it officially comes out, but shouldn't be too long.

I hope this wasn't boring, but I've never done this before and going through the process has been interesting.