Einstein's Energy

Energy equals mass times energy squared or e = mc^2. I think I can remember seeing that equation before I even knew what an exponent was. I didn't know what it meant, but it seemed important. Most everyone has seen it. My guess is that most, including students, don't really understand it, though.

Part of its beauty is not just that it is important, but you would think something this important would be a little bigger. It's really rather small as equations go.

Here are some fun facts about the equation.

• Giving off energy corresponds to a loss of mass. E = mc^2 tells us how much.
• The correct labels must be used. Being from Europe, Einstein used metric measurements.
• E is measured in Joules, m in kilograms, and c the speed of light, which is 300,000,000 meters per second.
• This formula was discovered by Einstein in 1905
• Chemical reactions, such as burning coal or wood are such that for the energy released, mass is lost, but it is almost immeasurable. Because of this, the formula wasn't physically shown to be true until 1932 - 27 years after Einstein's discovery.
• Einstein did not win a Nobel Prize for this or, for that matter, anything he did with relativity. He won it for the photoelectric effect.

There are a number on interesting problems students can work out that use this equation and a knowledge of scientific notation and rules of exponents. 

For example, The first atomic bomb released approximately 10^12 joules of energy. How much matter does that correspond to?

The matter in this case was Uranium-235. Using e = mc^2,

 10^12 = m(3x10^8)^2 gives approximately 0.000011 kilograms of mass.

Vitruvian Man II

 Last week I brought up some facts on ratios in the human body and how that relates to areas such as professional basketball. They weren't my facts, but from the book The Sports Gene.

This has to do with Leonardo da Vinci's sketch of The Vitruvian Man. That made me curious about that topic and how it got that for a name. I did a little research and here, what I think, are a few  are interesting tidbits on this topic.

• The drawing gets its name from Vitruvius, who was an accomplished architect in the Roman Empire, born approximately seventy years before Christ.
• Vitruvius made no drawings of his man, but described him.
• Several Italians including da Vinci made drawings based on Vistruvius' description, but not until 1,500 years later.
• Although others made drawings, da Vinci was the first to overlay the circle and square to show relationships.
• Leonardo was born in 1452 in Vinci, Italy - Thus "da Vinci".
• Leonardo's drawing was in one of his notebooks, which is why there is writing above and below the drawing.
• Even though it was on paper, the drawing still exists. It is located in Venice and is occasionally displayed.
• As mentioned in The Sports Gene, a person's wingspan is not always the same as the height (although mine is to the inch). If interested, there are several good class projects involving measurement and The Vitruvian Man on the internet. 
• Leonardo of course was famous for other works of art as well. I'm guessing that this along with The Last Supper and The Mona Lisa make him the artist with the most parodied works ever. (See below for a small sampling).

Vitruvian Man

I read an interesting book recently called, The Sport Gene, written by  David Epstein. Why do some excel in athletics and some don't - even with seemingly equal amounts of training? I always have liked to think that effort made all the difference. Anyone could be a great musician, a great high jumper, a great quarterback if one is just willing to put in the work.

Work is part of the equation for certain, but it turns out it's not the whole thing by any means. Some are just born with certain advantages.

Ted Williams was a great baseball player and some think maybe the best hitter of all-time. I'm sure he worked at hitting, but he didn't work at having 20/10 vision, which seemed to be a major help to him.

Muhammad Ali "reacted to light in 150 milliseconds, near the theoretical limit of human visual reaction time". Again, this is probable something innate and not developed.

The head of a athletic performance center said, "We've tested over ten thousand boys, and I've never seen a boy who was slow become fast."

What does it take to become an NBA player. One thing that helps your chances is height. The average NBA player is 15% taller than the average male. But it's more complicated than that.

The Vitruvian man is a famous drawing by Leonardo DaVinci. One can see that by Leonardo's placement of the man in the square is his assumption that a man's arm span is the same as his height. Is this always true? A class might try taking measurements and see if there is indeed a ratio of 1:1. Likely it will come quite close to that. I did my own measurement - exactly 1:1.

The book makes the point that in the 2010-11 season, there were only two players under that ratio. NBA players have an average ratio of 1.063:1. One of the current stars of the league is Anthony Davis. He is tall - 6 feet 9.75 inches, but his arm span is 7 feet 5.5 inches. That is a ratio of 1.095. He is tall, but effectively taller than his height would suggest.

Extra Points

The NFL regular season is over now. They did an experiment with extra points this year and it might be interesting to see how it turned out.

The NFL felt that extra points after touchdowns had become kind of boring. After scoring a touchdown, there is a kick from the two yard line worth one point which was pretty much always successful. During the 2013 seasons it was been successful 99.6% of the time. In 2014 it was successful 99.3% of the time. Not a sure thing, but pretty close.

Teams also had the option of going for two points by running or passing it in. That is harder to do, so, despite the lure of two points teams didn't usually opt for it unless it was near the end of the game and a team really needed those points.

To spice things up they decided to keep the two point option just as it was, but the kick had to made from the fifteen rather than the two yard line. Now that the seasons is over, we can look at how it turned out and what a team might strategically decide to do next year.

Starting from 2013 to the present season, since there were no changes, we would expect the two-points to stay about the same.

2013 - 33 for 69 - 47.8%
2014 - 28 for 59 - 47.4%
2015 - 45 for 94 - 47.8%

(I couldn't find a source that gave the stats for the entire league, so I had to add the team totals. I double checked, so I believe I'm correct on these numbers.)

While we would expect the success rate to stay about the same, it is kind of amazing to me that the percentages came out so close. We do see an increase in the number of tries last year, which is probably due to that fact that the one-point tries are not as automatic as they used to be.

One-point conversion success did go down this year, but not a lot.

2013 - 99.6%
2014 - 99.3%
2014 - 94.2%

So now, a question might be, "Should we just go for two every time now?" That boils down to what is our expected value for the various attempts.

2013 - One Point Attempts - 0.996x1 = 0.996. Two Points Attempts - 0.478x2 = 0.956.  Kick it.
2014 - One Point Attempts - 0.993x1 = 0.993. Two Points Attempts - 0.474x2 = 0.948.  Kick it.
2015 - One Point Attempts - 0.942x1 = 0.942. Two Points Attempts - 0.478x2 = 0.956.  Go for two.