The problem of intersecting cylinders

Santino G.

The other day, I ran into a pleasantly difficult math problem concerning intersecting cylinders, the solution of which involved quite a bit of imagination. It took some time until the key insight dawned on me, but after that, the math was very straightforward. The problem is from the second edition of “Calculus With Analytic Geometry” by Howard Anton; accordingly, all credit goes to him for the exercise which I reproduce below (the solution, however, is my own): In the tradition of Sal Khan, if you haven’t already, take a stab at the above problem before reading my solution. Soln. To … Continue reading The problem of intersecting cylinders

The physics governing bike riding

Santino G.

The warmth of spring, though currently intangible to our senses, is nearing closer, and with it, an environment favorable to the prospect of frolicking outdoors. While the list of soon-to-be-resumed activities is inexhaustible, one particular pastime has, for several weeks, maintained a relentless occupation in my mind: bicycle riding. Even now, with temperatures firmly set in the thirties, I’ve had a strong urge to get out the ol’ bike and take a ride about. I would act on this urge now, of course; alas, my bike hangs high up on the ceiling of my garage, imposing a steep activation energy … Continue reading The physics governing bike riding

An AP Physics student’s understanding of torque

Santino G.

The blog you are about to read (or perhaps quickly click away from after realizing there’s math involved) is merely the cumulative notes of my AP Physics 1 study. I therefore suggest that caution be taken when reading this material, as complete accuracy is not guaranteed. That said, I’m well-read and pedantic enough that I am fairly confident in this blog’s correctness. In the coming weeks, I plan to publish many essays like this, following the words of the late, great physicist Nicholas Wheeler: I cannot gain the feeling that I “understand” a subject until I have done my best … Continue reading An AP Physics student’s understanding of torque

Projects on my New Year’s to-do list

Santino G.

The Gregorian calendar is on the precipice of a new year, meaning it’s time for people to stay up until midnight, watch the ball drop, and, most importantly, declare their ambitious resolutions. While I don’t have any resolutions, per say, I do have a to-do list, filled with projects I need to start/finish. Nero Having now test-flown (/crashed) four iterations of my 3-channel radio-controlled (RC) plane, I am confident that I am nearing the optimal aircraft design: streamlined, lightweight, and powerful. The current plane in production is Nero, an aircraft that will be practically identical to Claudius, deviating only by … Continue reading Projects on my New Year’s to-do list

Regulatory hurdles in my model rocketry hobby

Santino G.

To regulars of this blog, long elaborations of radio-controlled (RC) plane failures ring a familiar tone. While much of my free time is indeed spent in such endeavors, there is an additional hobby of mine that I have barely discussed and yet has successfully consumed most (if not all) of my leftover leisure: model rocketry. This isn’t by accident, of course. There is very little to write about due to regulations preventing my procurement of a launch site, something that’s sort of necessary to have. Here, I collate some of the frustrating processes I have undergone in hopes that someone … Continue reading Regulatory hurdles in my model rocketry hobby

Orthogonal projections in linear algebra

Santino G.

In linear algebra, we are occasionally interested in resolving vectors into their “projections.” In contrast with components, however, which are determined with respect to the x- and y-axis, projections are determined with respect to an adjacent vector, one that shares the same initial point. This blog discusses the derivation of formulas that help us in finding these projections. Suppose we have two nonzero vectors u and v in 2-space. The vector u can be written as a sum of vectors, say, and : (i) where is constructed to be parallel to the adjacent vector v and is made to be … Continue reading Orthogonal projections in linear algebra

Conquering flight: The flight of Claudius (Part 10)

Santino G.

The temperatures have continued their gentle decline as planet Earth’s geographic north pole has strayed farther and farther from the sun. Nevertheless, the warmth provided by sweaters and experimental aircraft kept all of us comfortable that morning. Indeed, with the confidence I felt in the upcoming flight, I might have mistaken the season as spring. Embarking soon after sunrise, we arrived in step with the large morning attendance and immediately started to unload the newest iteration of the 3-channel RC plane. With its new weight-reducing features, Claudius was the lightest aircraft I had ever built. Some notable improvements were the … Continue reading Conquering flight: The flight of Claudius (Part 10)

Conquering flight: Claudius (Part 9)

Santino G.

One problem has pervaded all of my previous RC planes: weight. While I have been trying to minimize this one parameter in every plane since Augustus, it was only after I flew my third RC plane, Caligula, that I understood a more aggressive approach was necessary. This blog describes the construction of my fourth 3-channel RC plane, Claudius. Beginning with the fuselage, I maintained the same design used on the previous two; that is, foam board cut and folded so as to feature a square cross-section. It was, however, slightly shorter than Caligula’s. (This wasn’t an intentional decision; I simply … Continue reading Conquering flight: Claudius (Part 9)

Some physics pertaining to model rocketry

Santino G.

Despite my earnest pursuit of the model rocketry hobby, my efforts have been continuously undermined by the lack of one seemingly trivial thing: a launch site. This blog will not be focusing on this regulatory struggle (though, that would make a good post). Instead, I will collate some of the interesting physics I have learned regarding model rockets. The reader may rest assured, however, that my posts about my RC plane hobby will continue shortly. (Claudius is the next plane in construction and should be lighter than Caligula, Tiberius, and Augustus.) Center of mass (CM), center of gravity (CG), and … Continue reading Some physics pertaining to model rocketry

The fundamentals of Java

Santino G.

Being an object-oriented programming (OOP) language, Java has a substratum of classes and objects set as its foundation. This blog will briefly describe these two concepts. Classes In order to describe a real world object in our program, we start with a generalized version of it, a version with all of the attributes and behavior of the thing, but without any specific values. In Java, we call this uninitialized object a class. You can think of a class as a blueprint. For instance, when planning the construction of a neighborhood, a contractor will design a handful of designs for the … Continue reading The fundamentals of Java