### Newton's Cannonball

by: Lori, via the Exolab Blog

A deafening boom. A rush of air. The smell of gunpowder. And suddenly a cannonball was arcing through the air. This hypothetical cannonball, launched by Isaac Newton in a thought experiment, could be considered the shot-heard-'round-the-world heralding the arrival of the theory of gravitation.

Newton reasoned if one shot a cannonball off the top of a mountain at a relatively low velocity, it would travel some distance and fall back to Earth. Faster velocities would mean the ball travels farther before it hits the ground. But of course, Newton did not stop there. What if the radius of the ball’s trajectory curve matched the radius of the Earth? The ball must keep falling, but it would never reach the ground. Essentially, it would orbit. So technically, we are falling around the sun, while the moon falls around the Earth! This particular thought experiment that led to Newton’s theory of gravity was only made known after his death. He published very few works, though they were in high demand by some scholars; most of his writings were released only posthumously. But those he did publish, like the enduring Principia in 1687, sent waves through the scholarly community.
Though there were only a few hundred copies in print, Principia could be heard discussed in coffee-houses across England. It held so much information on so many different arenas of science and mathematics, that Marquis de l'Hopital declared of Newton, “Does he eat and drink and sleep? Is he like other men?”

Indeed, Newton was not like other men. As a teenager, he was constantly inventing curious contraptions, such as a mouse-powered miniature windmill, and doodling sketches of animals, people, and shapes on the walls of his boarding home. It says something of his brilliance that in grammar school, he was never taught natural philosophy and only basic forms of arithmetic and math, and yet approximately four years later he discovered calculus! He was raised in a relatively well-off economical situation, but his life was devoid of human connection. Raised by his grandmother, his father passed away before his birth, and his mother left her only son for a new husband many years her senior. In grammar school, the other boys were everything but friendly--likely put-off by Newton's unusual intellectual superiority. Lacking friends at such a young age, he devoted all mental faculties to the pursuit of sciences, and having been scorned by his peers, he carved his name into every school bench he occupied. Later in life he would suffer from several breakdowns, causing his contemporaries to question the soundness of his intellectual abilities. Newton was thus an enigma to those around him. His wildly creative nature was incomprehensible to many, leaving him quite alone.

Through all this, Newton found solace and inspiration in God. While working on Principia, he and one of his few friends, John Locke, would exchange letters, in which Newton would dissect various Biblical passages and their meaning. Newton’s view on the physics of the universe, similar to Kepler’s, were strongly connected to his devout, yet unorthodox Christianity. He would point to God as the answer to many cosmological curiosities, such as how "dark matter" (planets, rocks) was so distinctly separated from "light matter" (the sun, stars): "I do not think explicable by mere natural causes but am forced to ascribe it to the counsel and contrivance of a voluntary Agent."

Surviving through an emotionally turbulent childhood, Newton first studied mathematics until the Great Comet of 1680 turned his eyes heavenward.  Johannes Kepler's existing three laws of planetary motion provided a basis and reference for Newton to postulate his own three laws of universal motion.  Newton set off to discover the force which consistently drew objects downward, toward the center of the Earth. The theory of gravitation was, consistent with the famed legend, born out of a contemplation of the trajectory of a falling apple (though the apple never actually hit him on the head), which later spawned the cannonball theory. And, if gravity worked on the Earth, why shouldn’t it be applied to the moon and the rest of the cosmos? Newton declared that the heliocentric model was in fact correct, with one caveat. It was not the Earth that revolved around the Sun, nor the Sun that revolved around the Earth, but rather, both revolving around the system's center of mass. In his own words, "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World."

For all his eccentricity, Newton died quietly in his sleep in 1727. His equations for the motions of the planets and other celestial bodies are crucial foundations for exoplanetary science. In addition, his stipulation that components of a system orbit about a center of mass is the cornerstone of the radial velocity method of detecting planets. His crowning achievement is his theory of gravity, and if you were to fire a cannonball from the top of a mountain today, you’d find that the results would not have changed from Newton’s original postulations. However, almost 200 years later, gravity would be radically expanded to address high energies and speeds, permanently changing our perspectives on astrophysics by a German man with disheveled hair, who would come to be known as the father of modern physics: Albert Einstein.

“I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”
Isaac Newton

### On the Height of J.J. Barea

Dallas Mavericks point guard J.J. Barea standing between two very tall people (from: Picassa user photoasisphoto).

Congrats to the Dallas Mavericks, who beat the Miami Heat tonight in game six to win the NBA championship.

Okay, with that out of the way, just how tall is the busy-footed Maverick point guard J.J. Barea? He's listed as 6-foot on NBA.com, but no one, not even the sports casters, believes that he can possibly be that tall. He looks like a super-fast Hobbit out there. But could that just be relative scaling, with him standing next to a bunch of extremely tall people? People on Yahoo! Answers think so---I know because I've been Google searching "J.J. Barea Height" for the past 15 minutes.

So I decided to find a photo and settle the issue once and for all.

I then used the basketball as my metric. Wikipedia states that an NBA basketball is 29.5 inches in circumfe…

### Finding Blissful Clarity by Tuning Out

It's been a minute since I've posted here. My last post was back in April, so it has actually been something like 193,000 minutes, but I like how the kids say "it's been a minute," so I'll stick with that.
As I've said before, I use this space to work out the truths in my life. Writing is a valuable way of taking the non-linear jumble of thoughts in my head and linearizing them by putting them down on the page. In short, writing helps me figure things out. However, logical thinking is not the only way of knowing the world. Another way is to recognize, listen to, and trust one's emotions. Yes, emotions are important for figuring things out.
Back in April, when I last posted here, my emotions were largely characterized by fear, sadness, anger, frustration, confusion and despair. I say largely, because this is what I was feeling on large scales; the world outside of my immediate influence. On smaller scales, where my wife, children and friends reside, I…

### The Force is strong with this one...

Last night we were reviewing multiplication tables with Owen. The family fired off doublets of numbers and Owen confidently multiplied away. In the middle of the review Owen stopped and said, "I noticed something. 2 times 2 is 4. If you subtract 1 it's 3. That's equal to taking 2 and adding 1, and then taking 2 and subtracting 1, and multiplying. So 1 times 3 is 2 times 2 minus 1."

I have to admit, that I didn't quite get it at first. I asked him to repeat with another number and he did with six: "6 times 6 is 36. 36 minus 1 is 35. That's the same as 6-1 times 6+1, which is 35."

Ummmmm....wait. Huh? Lemme see...oh. OH! WOW! Owen figured out

x^2 - 1 = (x - 1) (x +1)

So $6 \times 8 = 7 \times 7 - 1 = (7-1) (7+1) = 48$. That's actually pretty handy!

You can see it in the image above. Look at the elements perpendicular to the diagonal. There's 48 bracketing 49, 35 bracketing 36, etc... After a bit more thought we…