Nerd Squared : A Saturn V Rocket Built By Harry Potter

I am a huge Harry Potter fan. When I was 12 or 13, I got a Harry Potter trivia game for Christmas. When I was asked what flavor popsicle Harry ate when he went to the zoo with the Dursleys, I not only knew the answer. I was able to look up the page in the book to prove to my brother that Harry had a lemon popsicle, not a cherry flavored popsicle as the trivia game incorrectly claimed. From that moment on, no one would play that game with me.

Now that I am studying math and physics, I was naturally interested in some of the math and physics behind Harry Potter. I was really intrigued with broomsticks. The other day, some information from a Harry Potter book caught my eye. In the Prisoner of Azkaban, the Firebolt broomstick's top speed is advertised as 150mph (or 67m/s) and can accelerate to that speed in 10 seconds.

Acceleration is a measurement of the change in velocity relative to the change in time. It is equal to the change in velocity divided by the change in time, so we can calculate that the acceleration of the broom is (67 m/s)/(10 sec) or 6.7m/s². I looked at Amazon's shipping information to find out how much the average broom weighs. A typical broom weighs about 1 pound, or 0.5kg. Daniel Radcliffe (the actor who plays Harry Potter) has a listed mass of 53kg.

Since Force = Mass * Acceleration, we know that the amount of thrust produced by one Firebolt is equal to (6.7m/s²) * (53.5kg), or around 360 Newtons.

While this is an interesting factoid, knowing the amount of thrust that something produces doesn't really mean a whole lot without any context. I wanted to compare the thrust to the amount of thrust that other vehicles can create to really get a sense of how powerful these broomsticks are.

What I found would probably surprise no one. Cars, boats, planes, and rockets can all produce much more thrust than a broomstick. But a broomstick weighs so much less than these other objects! If we had the same mass of broomsticks as the mass of whatever is powering the craft, I'm sure the broomsticks would dominate! So let's look at that.

I wanted to look at the most extreme example I can find: the Saturn V Rocket.

Weighing in at a whopping 5 million pounds, this world heavy weight champion of spaceflight was used to fly the Apollo missions to the moon. This rocket used 2,160,000 kg of liquid oxygen and liquid hydrogen as rocket fuel and produced a thrust of 7.5 million pounds (or 3.4*10⁷ Newtons). The first stage of the engine burned for 168 seconds, at which time the rocket was at an altitude of 67,000 meters and was moving 2,300 m/s.

OK, there are many different ways to compare the NASA model and the HP model. If we just straight up traded the mass of fuel for the mass of broomsticks, we would have around 4,320,000 broomsticks. At the rate of 360 Newtons/broomstick, this would create 1.6*10⁹ Newtons of thrust, easily surpassing the liquid oxygen and hydrogen.

Or we could figure out how many brooms we would need to create the amount of thrust needed. We still have 360 Newtons/broom, so 3.4*10⁷ Newtons requires 95,000 brooms. Cool!

But this doesn't really tell the full story. One of the main reasons rockets are so heavy is because they have to carry their fuel with them. They start out extra heavy and end up with hardly any mass left. But if we used broomsticks, that mass would not change. NASA is all about efficiency, and any good Quidditch player doesn't want to waste any broomsticks. So we want to figure out the minimum number of brooms to reach the stars.

Remember that brother that refused to play Harry Potter trivia that I talked about above? He's now a rocket scientist. He got his undergraduate and graduate degrees in Aerospace and Mechanical Engineering at the University of Michigan (Go Blue!) and now works as an engineer at Boeing, so I naturally tagged him in for this fight.

I'd recommend using a Fourier Transformation then finishing it off with a Suplex.

He was very intrigued. You see, the really tricky part about rockets is that they require a LOT of fuel. They are constantly burning a massive amount of fuel, so the mass you end up with is very different from the mass you start with. My brother said that engineers normally deal with Tsiolkovsky's equation, which is:

where  is the velocity of the exhaust,  is the initial mass (both rocket and fuel),  is the final mass, and  is the maximum change in velocity of the craft. Since we are using magic to power our rockets, there is no change in mass.

This means that m/m above is equal to 1, and as we all know after looking at logarithmic functions, the logarithm of 1 is equal to 0. Therefore according to Tsiolkovsky, we cannot move.

My brother told me that since rockets are so aerodynamic and stable, we don't really need to worry about the drag force acting on the rocket. Rockets are specifically engineered to minimize the air resistance, which means that the drag force is significantly less than the amount of thrust produced by the rocket.

This is pretty much word for word how I asked my brother for help.

So this problem breaks down to a simpler-than-you-would-expect-for-rocket-science-meets-magic scenario. Since we don't have to worry about changing mass or drag forces, we only have 2 forces acting on the rocket: gravity and thrust. We know that these two forces added together equal the net force, and the net force needs to be great enough to get the rocket up to escape velocity (which is 11,200 m/s).

Force_Net = Force_Thrust - Force_Gravity

mass*v_esc/t = (360 Newtons/Broom)*(n brooms) - mg

(120,000+n/2 kg)(11,200 m/s)/(168 sec) = (360 N/broom)n - (120,000+n/2 kg)(9.81 m/s²)

33.3n + 8*10³ = 360n - 4.9n - 1.2*10⁶

322n = 1.208*10⁶

n = 28,500 brooms

So it would take about 28,500 Firebolt broomsticks to fly the Saturn V rocket into outer space. This seems like a lot, but is it?

The Saturn V rocket uses about 90,000 kg of liquid hydrogen, 1,800,000 kg of liquid oxygen, and 700,000 kg of RP-1 (rocket propellant 1, similar to kerosene). This fuel costs around $800,000 today. Meanwhile, the Firebolt broomstick cost around 1000 galleons in the Harry Potter books in 1993. This converts to about $10,000 in 1993 or $16,500 in 2014. This means that the total cost of the brooms would be 28,500*$16,500, or about $470 million.

Unfortunately this method of space travel would not save us money until about our 590th trip into space. While it might start getting more cost effective if we started sending spaceships farther out into space, for the immediate future it would not save us money. I say we go for it even though it costs more. I think half a billion dollars is a fair price to kick the laws of nature in the face!

Archimedes: Doing more math in 200BC than any of us can today

Most of you probably have heard of Archimedes of Syracuse. He was a Greek mathematician and (possibly mad) scientist alive in the 200B.C.'s. I like to call him the Corn Syrup Mathematician™ because, like corn syrup, you would be astounded to believe how many things people use today that contain something that Archimedes worked with.

Fun fact. Much of what we know about Archimedes was recorded by Vitruvius. I wonder if
we could get Morgan Freeman to narrate some of Archimedes work...

One of the really cool things about Archimedes is that a lot of the topics he worked on make sense. Look at other famous mathematicians and scientists. Does anyone think that what Einstein talked about truly makes sense? Does anyone look at the work of Heisenberg or Schrodinger and think "Oh, now that you put it like that, I totally get it"? In no way am I saying those scientists are incorrect, but the scientific breakthroughs they made are hard to conceptualize. Archimedes on the other hand is a great example to show to people that you can make great insights simply by using common sense and prior knowledge.

I honestly cannot tell if this is the most insightful thing I've heard today or useless.

Archimedes discovered many principles dealing with fluid displacement, buoyancy, and density. He built an Archimedes Screw, something that is still used today. He designed many different kind of siege warfare machines, and supposedly burned a fleet of ships using mirrors.

He also worked in mathematics. He gave an approximation of the square root of 3 (which was accurate to the hundred thousandths decimal). He calculated the area contained by a parabola and a straight line. He calculated the areas of triangles and parallelograms. He approximated π.

Coincidence? I think not.

The work he was most proud of looked at a sphere inside a cylinder where the height of the cylinder and the diameter of the end of the cylinder was equal to the diameter of the sphere. (I want to note here that at this time in history, calculus did not exist. Archimedes was able to accomplish all of this without integrals, infinite series, or limits. Or if he did use those concepts, he had to discover those concepts himself.) Archimedes actually requested that after he died, instead of a gravestone, he wanted a carving of the following figure placed over him.

Yeah, instead of an epitaph, could you just carve my 9th grade
essay on "Haroun and the Sea of Stories" onto a marble slate?

He first calculated the area of a circle by looking at a circle made up of 6 equilateral triangles. He realized that if you put the triangles next to each other, the the length is slightly less than the circumference and the height of reach triangle is slightly less than r. As you increase the number of triangles, the height of the triangles gets closer and closer to r while the length gets closer and closer to the circumference.

Area_Circle = (Area of 1 Triangle) * (Number of Triangles)

A_Circle = (1/2 * Circumference/n * r) * n

A_Circle = 1/2 * C * r

Since C = 2πr, we know that

A_Circle = 1/2 * 2πr * r

A_Circle = πr²

Now Archimedes thought about the surface area of the sphere. He decided to first start by looking at the surface area of a section of the sphere, defined by a section of the sphere cut off by a plane slicing through the sphere.

Archimedes claimed that the surface area of the section of the sphere that was sliced off is equal to the area of a circle with radius equal to the distance from the perimeter of the intersection to the top of the center of the circular area.

Since we are looking at the surface area, let's imagine that the sphere is empty. It is just a shell that outlines the sphere. For reasons that will be clear very soon, let's think of the sphere as the empty casing around a spherical roast beef at Arby's.

[Insert Product Placement Here]

I chose a hunk of roast beef so we can imagine slicing it with a meat slicer. If we were to keep slicing the meat thinner and thinner, we could imagine slicing a 2-Dimensional slice of meat. But remember that our special roast beef is just a hollow shell, so these 2-Dimensional slices are just circles.

"Isosceles. You know, I love the name Isosceles. If I had a kid,
I would name him Isosceles. Isosceles Kramer." - Cosmo Kramer

If we were to take all of these circles, cut them, and lay them out straight so that their ends were level, we would see something like this (this is very similar to the situation we looked at above).

As the slices get thinner, the number of slices increases and the picture above starts to look like a triangle.

Area_{Cross Section} = 1/2 * Circumference * Radius

A_CS = 1/2 * C * r

Since C = 2πr, we know that

A_CS = 1/2 * 2πr * r

A_CS = πr²

Therefore, we know that the surface area of the portion of the sphere that is cut off by a plane slicing through the sphere is equal to the area of the circle defined by the intersection of the surface of the sphere and the plane. Now if we imagine that the plane cutting the sphere goes through the center point of the sphere, the radius of the intersection circle is equal to the distance from point (0, 0, r) to 

(r, 0, 0). This distance is equal to √2r. Since this only calculates the surface area of the top half of the sphere, the surface area of the sphere is twice the area of the circle with radius √2r.

Surface Area_Sphere = 2*π*Radius²

SA_S = 2*π*(√2r)²

SA_S = 4πr²

Now at this point, Archimedes kind of goes off the deep end. He started talking about approximating the volume of a sphere using pyramids. I looked at how he calculated the volume of a sphere and had the following reaction.

I don't quite understand it myself yet. There is a strong possibility that I write another blog post about more of Archimedes work, but for now