Finding Treasure

In class, we've talked about how the location of a buried treasure can be described by using a displacement vector. If a treasure hunter knew where to start (the origin) and the displacement vector, he or she would be able to find the treasure.

What about other coordinate systems? Suppose instead of a displacement vector, the treasure hunter knew exactly how far the treasure was away from two fixed points. If this were the case, would he or she be able to dig up the treasure on the first try? How many possible locations for the treasure would there be?

Pythagorean Triples

Photo by Gavin Keefe Schaefer

As we've been studying vector addition in class, I've been coming up with examples to use on the board. I usually deliberately choose the addends so that the magnitude of the resultant will be a whole number. How do I do this?

I choose the addends so they are two members of a pythagorean triple--three integers that are sides of a right triangle. Have you memorized any pythagorean triples?

1. Is there a pattern or formula that can be used to generate pythagorean triples?
2. Is there a size limitation? Is there such a thing as the largest phythagorean triple or the last pythagorean triple?
3. Do you think there are any pythagorean triples in the photo at the top of the post?

Compass Confusion

Photo by Calsidyrose.

When talking about directions, high school students are often presented with conflicting coordinate systems. In physics (and engineering, navigating, and surveying) we assign north to be 0°, and we then work around the compass in a clockwise manner. In contrast, mathematicians assign the positive x-axis to be 0° and then proceed in the counterclockwise direction.


Physics teachers (and physics textbook authors) are faced with a dilemma: should we stick with the same coordinate system used in the students' math books, or should we teach students the coordinate system that they are likely to see on a real-world compass?

1. There is no "right" answer to this dilemma, but what do you think is the best solution?
2. Is there an angle where a student using the "physics" coordinate system and a student using the "math" coordinate system will agree on the same value? In other words, which angle(s) is the same in both coordinate systems?
3. Many people have and use GPS navigation devices. These devices are capable of reporting the current course heading. If you own or have access to such a device, experiment with it by moving in a known direction. Which coordinate system does your device use?

Vector Addition

In class, we talked about how a vector is a mathematical quantity that has both a magnitude and a direction. Graphically, we can represent a vector as an arrow--the magnitude is indicated by the arrow's length, and the direction is represented by where the arrow points.

In physics, we will find it necessary to carry out mathematical operations with vectors. One can understand how vector addition works by using graphical representations (arrows). In class, we discussed two common ways to graphically add two vectors--the "tip to tail" method and the parallelogram method. Play around with the interactive vector addition tool at http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html.

1. Which method of vector addition do you prefer?
2. Is vector addition commutative, e.g., if A and B are vectors, is A + B = B + A always true?

Distributed Computing

Can you imagine doing an experiment so large in scope that a real limit is that it would take a modern computer thousands of years just to process the data? In fact, there are many experiments that have this exact problem.

One solution to the problem is to use a very expensive and/or specialized (super)computer. This is often not possible. Many scientists and computer scientists have taken an alternate (and creative) approach. They tackle these huge data sets with a technique called distributed computing. Essentially, they use the internet to share the data in small portions to thousands of normal computers around the globe. Computer users allow their computers to process data while they would otherwise be idle. Read more about the technique at wikipedia.com.

Many of these experiments are physics projects. Perhaps one of the most famous examples of distributed computing is SETI@home.

1. Research some current examples where distributed computing is being used to solve a problem. What examples are the most interesting to you?
2. Would you be willing to allow your home computer to be used for distributed computing? Why or why not?
3. Do you think our school should allow all of its computers to participate in a distributed computing project? Are there any costs associated with it? Would it cost the school money or would it essentially be free?