Galileo Galilei and Parabolic Motion Short Story
Manvir and I did a short story geared towards secondary math students where the story incorporates modern day football and a historical lesson on parabolic motion.
Thinking about the history of math, I think one of the centric themes of this course was to be able to tie in history of math into practical lessons to teach your secondary students. Thus, we looked at the maths that go into throwing a football, and how fundamental parabolic trajectory and projectile motion are to the sport. Therefore, our idea was to use a possible scenario where the quarterback wants to throw further independent of strength, and what parameters of his throw he can change (as per parabolic motion) to increase distance in the horizontal direction. We did a deep dive into how Galileo proved the fact horizontal and vertical velocities are independent components, and what parameters affect each vector.
After our research, we decided a short story/play would be a very interesting medium!
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The lights are on, and the field is shining and stadium is full here in Foxborough, Massachusetts. It is the Sunday night game for the New England Patriots, and coach Bill Bellicheck was watching the players closely. Tonight, his eye was especially keen on star quarterback Tom Brady. He had just taken a massive hit, and was coming off the field looking a bit dazed.
Coach Bellicheck was scared, walking over to Brady.
“Tom, you good? I don’t think you are in any position to continue this game”
“I don’t know what it is, I can’t really even say where I am right now” mumbled Brady.
This calls for extreme measures thought Bellicheck. He knew he could only do this once, and with the playoff spot on the line, it was time to take out this card.
“Well, Jarrett, you are up” called Bellicheck to the backup quarterback
“Before you go, I want you to go back to the locker room and meet someone.”
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The Patriot’s locker room was completely empty. No rock music playing, no physical trainers running around. Everyone was on the field, as this was the biggest game of the year.
A white man in his late 70’s with a long grey beard in a black robe with a white collar was sitting right next to Jarret’s locker. He was sitting as if he had been waiting for a thousand years.
“Jarrett, my name is Galileo Galilei, and I have time-travelled here to perfect your throw by giving you the wisdom of parabolic motion”
“Young man, there is a reason why Tom Brady can throw so well, I once had this same conversation with him, and thus I am here to pass you on the same knowledge. Tom Brady knows the maths and physics behind the motion of a football.
“You see, I discovered the maths of parabolic trajectory when I was under house arrest. You see I introduced a theory that the Sun was the center of the solar system known as heliocentrism which the Church opposed since they were preaching geocentrism. The theory that the Earth was at the center of the solar system.”
Jarrett, shocked by the events unraveling in front of his eyes, abruptly interrupted and the first out of his mouth were “Wait, why were you on house arrest?”
Galileo explained “Back in the day, the church was law and if you opposed it you were sent straight to prison. They didn’t like the fact that my findings indicated that we were not the center of the solar system, the sun was. Trust me Jarret, I got off easy with house arrest”
Jarrett, still confused by what was going on, asked “Ok, but what does this have to do anything with throwing a football?”
Galileo answered with excitement, “Jarett, throwing a nice spiral 40 yards down the field is an art but simultaneously it is also a science. I spent the last few years of my life discovering and mastering parabolic motion. I’ve sat down with Bill and studied your game film. We need to go back to the drawing board and improve your throwing motion.”
Jarret, intrigued but hesitant, said “Galileo, sure you may know the science but football wasn’t even a thing when you were alive. We got the best trainers here in foxborough and we’ve been working every day.” Jarret glanced over at Bill,”You’ve had a lot of quirky ideas, not too sure about this.”
“You see it all begins in 1590 when I had proven mathematically that in theory a ball on a level plane will be set in motion by any little force. When I was studying parabolic motion, I placed an inclined plane on a table and put a curved piece at the bottom to deflect an inked bronze ball into a horizontal direction. When it rolled off the table, and hit the floor it left a small mark where I was able to measure the horizontal distance travelled and the vertical distance.
“Then I decided to test for what would happen if rather than just rolling along a horizontal plane, if the ball just dropped vertically after coming down the inclined plane, (removing the horizontally deflecting curve)
My hypothesis was that if the horizontal and vertical motions were independent, the ball would still move horizontally with uniform constant speed but the gravity would pull it down quicker to the elapsed whereas if there was a curve the ball would still be travelling.
And indeed, the ball still had horizontal motion!”
“Through my precise measurements, I was able to determine that the vertical displacement of a falling body was related to the square of the time. This relationship can be expressed in the equation
s = 1/2 gt2
“So, when you throw the ball up Jarrett, there are really two directions and velocities. I confirmed that these are both independent, you will have your vertical component and a horizontal component which both we can represent using vectors. Through this we can determine a projectile motion to have a parabolic trajectory and using parabola math which I did not invent. The Greek mathematician, Menaehchmus, was the one to come with the mathematics behind a parabola when he was attempting to solve the problem of the doubling cube.
“Galileo, so what do you want me to do?”
“Coach Bellicheck has told me you are strong, but the distance of your throws is very small. This is because the distance of the ball can be measured can be derived using the formula sin(2theta)*v^2/g.
You see the sine function reaches the maximum value of 1 at an angle of 45 degrees. The key Jarrett when you are throwing because of parabolic motion is to throw at that angle.”
“I had never been taught such knowledge, thank you Galileo.”
Jarrett got substituted into the game, and it was the fourth quarter and the Patriots were down by three with 60 seconds left in the game. The Seahawks had fumbled the ball, and now the Patriots had the ball and it was Jarrett’s time to shine. On his first touch of the ball, he angled his arm at a perfect 45 degree angle and threw the longest yard NFL touchdown of the year, and the Patriots won the game.
References
- Drake, S., & MacLachlan, J. (1975). Galileo's Discovery of the Parabolic Trajectory. Scientific American, 232(3), 102-111. Retrieved from www.jstor.org/stable/24949756
- Wohlwill, E. (2001). The discovery of the parabolic shape of the projectile trajectory. Science in Context, 14(S1), 375-410. doi:10.1017/S0269889701000448
Great work, Jovan and Manvir! I really appreciate your creativity in bringing together some interesting ideas from the history of math and physics with your interest in sport. A play like this would go over very well with the sports fans in a secondary school class!
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