Georgetown Day School

Ed Eckel and Deb Maresh

PreCalculus and Honors Physics - High School Level

Syllabus

Goal: Structuring understanding

Note: These courses are separate but coordinated; different student population will be found in each. Of the 60 students served, approximately 10 will be in both classes.

Course Descriptions:

PreCalculus: The students entering PreCalculus have already studied a considerable amount of trigonometry. Very few topics in trigonometry will be new to them. The students will review trigonometry and quickly move on to modeling data, statistics, probability, conics, graphs of functions, and the definition of a limit. The order that the topics will be covered is not set in stone as I teach only two of the four PreCalculus classes, and students changing their schedule could become a big issue. Some coordination must be done with the other PreCalculus teacher.

Honors Physics: The students in Honors Physics span grades 10-12. Some will be in Pre-Calculus and others in the Algebra 2/Triginometry offering. The course will cover Classical Mechanics, Thermodynamics, Electricity and Magnetism and select from Optics, Fluids, and Modern Physics depending on time available and interest. An essential feature will be the use of experiment to inform conceptual understanding and conceptual understanding to dictate the mathematical description of phenomena studied.

PreCalculus

Trigonometry (incorporate programming on the calculator)

Periodic functions/the six trig functions (review)

*x=cost, y=sint*

Graphs/modeling (review)

Properties/Identities/Combos (half review)

Rotary motion/inverses/applications (half review)

Triangles/vectors (review)

Graphs of functions(polynomials and rational)/Limits/Derivatives

Quick recap of the graphs and properties of specific functions

Graphs

Notion of limits

^{th} grader?
An 8^{th} grader?

Definition of a derivative

Fitting Functions to Data

Finding patterns in data

Scattered data

Lends itself well to a labs

"Hook up" with Ed’s 3^{rd} period class

Statistics/Probability

Discuss statistics found in magazines.

One variable

Combinations/permutations/random variable

Possible labs (? that aren’t trivial ?)

Conics

Graphs

Focus/directrix/center/etc

Rotation of axes

(?)Construct a paraboloid - find project from NCTM Conference

Visit www.cpo.com/weblabs/projecta.htm#Parabola

Polar Coordinates/Parametric Equations

Lab with position/distance and time

Complex Numbers/Series

Ball bouncing idea from class 6/21

Motion detector/CBL’s

Honors Physics

1. Newton and Aristotle: What Is Change

The purpose of this unit is to explore the Aristotelian and Newtonian worldviews. We will
observe and graphically represent several motions yielding various functional forms and
thus different shapes on position time graphs. This unit will be used to introduce the CBL
and MBL acquisition systems. Both position-position and position-time graphs will be used.

**Topics:**

Ball falling

Coffee filters falling

Ball bouncing

Simple Harmonic Motion (pendulum and spring)

2. Modeling

This unit will teach/review unit circle trig. We will start with the SHM from Unit 1 and
explicitly relate it to circular motion, describe the unit circle and define the trig
functions sine, cosine, and tangent in the mathematical way. Coordinate systems will be
introduced. The use of right triangle trig to move between coordinate representations will
be introduced. The essential feature is the position of an object does not change during a
coordinate transform, only the description of the position. Finally, we will look at the
relative value of a position-position plot (snapshot of space frozen in time) and the
position-time plot. We will introduce parameterization as a means of creating a
"movie" of the motion. We will restrict the "movie" to one dimension
as ‘an artifact’ of making 2-D drawings with one axis devoted to time. Care must
be used to avoid the impression that a graph is ‘a picture’ of the motion.

Finally, we will explore using the spreadsheet to generate functions that yield graphs matching the data collected. We will link closely with math in the analysis of the features of the functions generated.

**Topics:**

Periodicity and the Unit Circle

Describing where something is

Trig functions and unit circle

Trig functions and similarity

Coordinate systems

Rectangular

Cylindrical

Spherical

Parameterization

Using an equation to represent the data

Use of Spreadsheet

3. Kinematics in 1-D: mass doesn’t matter!

Galileo’s Experiment will be repeated using ramps at different slopes, measuring
distance traveled and time to travel. Graphical representations created by all lab groups
will be used to search for patterns in the data. We will work with the math class in
developing functions that represent the rate of change in the data. We analyze the
functions and then repeat the process until we find something which does not change. We
ask: is a general pattern evident and how might we describe it? What assumptions are
buried in the abstracted patterns (standard kinematics formulas, definitions of speed and
acceleration and, in particular, the mass of the object is not accounted for).

**Topics:**

Galileo’s Experiment

Patterns in the data

Data Mining: Modeling, Slopes, and Re-representation.

Patterns in the Representations

4. Vectors: Fully Representing Pertinent Information

We consider the Galileo experiment and ask if the ramps had a part to play. We find that
the pattern of constant acceleration is robust from small to vertical slopes. We note the
slope of the ramp contributes a magnitude effect. We consider the generality of the
abstracted patterns. We ask about motions with superficially different character: the
tossed ball. We consider direction in addition to magnitude of motion. We investigate the
relation between horizontal velocity and vertical acceleration. We work with the math
class to test our understanding on stomp rocket flights (predict, test, evaluate.)

**Topics:**

Magnitude and Direction

Projectile Motion

Hypothesis: cause and effect between vertical acceleration and horizontal velocity

How Far I go depends on how far I fall!

Stomp Rockets

5. Newtonian View and Newton’s Laws: yes, mass
matters!

The stomp rockets reveal some problems: flights are neither as far nor as straight as
expected. We ask about the cause and effect nature of acceleration. What causes
acceleration and what influences its effect?

**Topics:**

First Law

Inertia is NOT a force

Second Law

Meaning of multiplication

‘Sum’ means all: FBD

‘Net’ means a single effect: acceleration

Cause and effect: knowability

Third Law

The difference between walking on water and walking in space

**Interregnum** (essay and reflection)

Time, Change, and the Ability to Understand: Forces and Nature

An Interregnum is an ancient Latin term for a pause. For me, it carries the connotation of reflection. Thus, we pause to reflect on what we have discovered, to gather to ourselves the critical elements of our investigations. This is a critical part of the course. We will spend time discussing the implications of Newton’s Laws on the worldview of the Physicist. We will examine a quote from Wittgenstein about the ability of a proof to change and indeed organize the way we perceive the world. We challenge the view from the concern of ideological rigidity. We find we must continue the journey, by looking at the implications of the definition of force.

6. Forces through time and displacement

As we study forces and the effect of the application of a force on a mass, we see that a
particular force is not required to produce a particular effect. In fact, it is the time
over which the force acts that seems to control things. That is, small force acting for a
long time may produce the same effect as a large force acting for a short time. We study
this through experimental observation, consideration of what the meaning of various
mathematical combinations of force and time. We examine the robustness of our conclusions
by looking for effects associated with the vector nature of force. Finally, we arrive at a
conservation law.

We proceed to consider the idea that forces acting for a finite and non-vanishing period of time also must act on the body after movement has been induced. That is, the point of application of the force moves while the force continues to be applied. We begin with simple observations about the application of a known force over different distances, move to a theoretical discussion, and then revisit the idea of conservation.

**Topics:**

Momentum

Impulse

Collisions

Decay (1 and 2-d)

Conservation

Energy

Kinetic Energy

Potential Energy

Work-Energy Theorem

Weak Form of Conservation of Energy

Non-Conservation under the weak form: friction and temperature increase

**Interregnum: A Mental Map**

Students produce a map explicitly revealing assumptions and constraints in linear
dynamics. A paper detailing the differences and advantages of Newtonian and Aristotelian
view is produced and shared. Each student makes written comments on the writing of the
others.

7. Project: Extending to Angular Dynamics

This is a group project in which the students are called on to use their maps to build a
kinematics and dynamics for rotational motion. This is expected to be hard. Worksheets and
seeds of lab ideas will be provided but student suggestions for investigations will be
avidly sought.

**Topics:**

Review of Angular variables

Angular kinematics

Moment of inertia (oops, the distribution of matter does matter, a lot!)

Torque (Where the cause happens is nice to know)

Angular momentum

Rotational KE

Conservation of Energy (weak) made Stronger

Hoop and disk races

8. Are all forces contact forces: Gravitation and
Electrostatic

This begins a fundamentally different section of the course. We look for ways of
understanding the motion of the moon about the earth. We look for confirmation of the
Keplerian reduction of Brahe’s data. We look for evidence of other phenomena that
behave as a non-contact agent of change. We look at abstractions associated with the study
of electro-magnetism but equally well applied to any causal agent.

**Topics:**

Newton’s Law of Gravitation

Kepler’s Laws as Empirical Construction

Kepler’s Laws as Theoretical Construct

Coulomb’s Law

Gauss’s Law

Work and energy

Potential and field

9. Kinetics: discovering complexity through theoretical consistency

**Topics:**

Model of matter

Phases of matter

Appearance of energy (temperature and heat flow)

Interactions (Pressure, volume)

10. Thermodynamics

**Topics:**

State variables

Intrinsic and extrinsic variables

First Law of Thermodynamics: absolute conservation of energy

Carnot Engines

Second Law of Thermodynamics: Modeling the Reality of Absoluteness

Entropy and the Reality of Puritan Ethics

11. Circuits

**Topics:**

Elements (batteries, wires, and extraction devices)

Relate to Carnot’s thoughts

Potential and current

Kirchhoff’s Laws

Capacitors

12. Magnetism

**Topics:**

The changing electric field due to a moving point source

The detailed atomic structure of naturally magnetic materials

Catalogue ways to make a magnet display the effect of external force on it

Ampere’s Law

Changing Magnetic Fields

Faraday’s Law

Magnetism and work

Solenoid

If we have time and possibly as small group study projects:

Optics

Fluids

Modern (as done by Zollman at UNL)

Next syllabus

Syllabi index

Introduction

7/30/99