Original Math Notes
All Seasons
Episode 402: Velocity--Explore the Math
Episode 402: Velocity
Interactive Computations
Use the free Wolfram Mathematica Player to interact with the math behind NUMB3RS.
Scene 4:
                    DON
          A car made of stolen parts from
          around the country drives straight
          into a coffee shop.

                    CHARLIE
          The CHP accident analysis will tell
          you how it happened.

                    DON
          But I want to know why.

                    CHARLIE
          That's tricky -- finding motive in
          skid marks and trajectories.

Projectile Motion »

The Moon has less gravity than the Earth, so a dropped or thrown object takes longer to land. In this Demonstration, the times and trajectories of hurled objects are plotted for various planets.
Scene 9:
                    AMITA
          I see you're reading Dr. Preskill's
          paper on quantum particles and
          multi-dimensionality.

Wavefunctions of Identical Particles »

Explore the quantum mechanical probability density for two particles in a one-dimensional square well with infinite walls.
Scene 10:
Galinski goes to a computer, punches up a screen. Detailed
computer graphic of a Mars Rover, then a car, then a truck --

                    GALINSKI
          My own computer modeling software --
          analyzes anything with an engine.

Radial Engine »

This Demonstration shows a working model of a four-stroke radial engine. Unlike a straight cylinder engine, the cylinders are connected to the crankshaft via a single hub with a master-and-articulating-rods assembly. This engine configuration was very popular in WWII aircraft. For smooth firing order, most four-stroke radial engines have an odd number of cylinders.
Scene 10:
                     GALINSKI
         Mechanical engineering -- the
         poetry of matter and energy, of
         metal and power. The more
         complicated it is, the more
         beautiful it becomes.

                     CHARLIE
         We need to know if a certain
         vehicle's trajectory was accidental
         or intended.

Rolling Disk »

This Demonstration shows snapshots of a disk rolling on a plane. For the calculation, it is assumed that the disk has infinitesimal thickness, rolls without friction, and does not slide or slip. A rolling disk is one of the simplest examples of a nonholonomic system. It is completely integrable, and as a result, the sequence of snapshots form nonchaotic sequences exhibiting a certain symmetry.
Scene 15:
                    CHARLIE
          Computer simulations are
          mathematical representations of the
          real world. Think of Newton --

          Newton observed a real world event-
          The APPLE falls from the tree.

          And he created a mathematical model
          to represent the phenomenon-

          That model could be extrapolated to
          predict the orbits of planets-

Orbital Elements »

Use this Demonstration to explore the five parameters which characterize orbits about a central mass.
Scene 19:
                    CHARLIE
          It's a phenomenon related to
          centripetal force --

Rhombic Drive for Speed Governor »

Four rods are attached to each other with hinges at the vertices of a rhombus. Two of the sides are extended, with balls attached at their ends. The top vertex has a constant height. As you rotate the whole assembly, the balls move upward due to centripetal force.
Scene 19:
                    CHARLIE
          The further from the center of the
          circle, the greater the distance
          traveled.

Circumference of a Circle »

The circumference of a unit circle is 2π, or about 6.28.
Scene 20:
The APPLE stops mid-air.  GRAPHIC overlay displays Newton's
equations of motion and the law of universal gravitation.

                     CHARLIE (O.S.) (cont'd)
         That model could be extrapolated to
         predict the orbits of planets-

The SOLAR SYSTEM -- planets in motion around the sun.

Orbits Around and Through a Sphere »

This Demonstration shows the orbit of a test mass in the gravitational field of a homogeneous sphere. Assume that the test mass can penetrate the sphere without frictional forces being exerted. Inside the sphere, the gravitational force forms a harmonic oscillator and outside the sphere, it has a pure Coulomb potential. In both cases, the orbits are ellipses. In case the orbit has segments both inside and outside the sphere, rosette orbits are obtained.
Scene 37:
                    GALINSKI
          And that, folks, is geek speak for
          "bingo."

Bingo Card Generator »

In the game of Bingo, players must be the first to meet various spot-filling criteria: a row, column, or diagonal; a "T"; an "H"; all spots filled, etc. The most common game is to be the first to fill five spots in a straight line.
 
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