To Paul about Physics C without Calculus First.
1.
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4.
5.
Except in 1 in 30 cases, all my Physics C students have had an Honors Physics first.
In Honors Physics we do a great deal of graphical Calculus, finding slopes and areas.
About 20% of my APC kids have completed Calculus prior to their senior year.
About 60% are taking Calculus at the same time as AP C.
About 20% are not taking any Calculus class during HS.
Since 80% of my students in APC have a need for Calculus skills prior
to their seeing the same skills in their Calculus course or because
they will not see Calculus, I teach the following over a 3 to 4 week
period (note - my class is an every other day class and are 90 minutes
each – standard A/B block):
1. Actually derive using “wave the hand limits” the polynomial
derivatives.
2. Derive Sum, Difference, Product, Quotient and Chain rules.
3. Derive the slope of Sin and Cos and nicely get to show that
sinx/x = 1 and cosx/x = 0 . I also make a point of sinx<x<tanx
all approach zero at essentially the same rate. Trig functions
are useful once in awhile, mainly to “prove” that Ac = V^2/R
without using the rotating vector approximation and since I also
want to teach magnetism it is useful in direct integration of
some of the harder integrals(See way below for a caveat).
4. I also teach e^x and lnx and logx derivatives and integrals. In
fact, these are the ones that APC seems to use the most. As we
have seen several times in the past several years including the
strange year when it was on the test twice.
5. I also do a not so small lab where they deal graphically and via
Calculus with some data to see that it actually does work.
Whenever we do motion experiments leading to acceleration I have them
plot only the position graph and then do an appropriate curve fit and
then do a double derivative. Sometimes I encourge my stronger students
to integrate the Fa=Kv^2 which they have found in doing the coffee
filter drop lab to determine predicted velocity and position. This is
actually more difficult than using the Fa=kv equation that APC usually
tests.
I insert Calculus as much as is possible simply to reinforce their
initial learning or to give further examples of the material they are
learning in Calculus. It probably does not hurt that I was initially
hired as a Math teacher and taught Pre-Calculus and Calculus for about
15 years. I have taught Calculus, Pre-Calculus, AP Computer Science,
Physics and APC during my career. The longest stretches were Physics
and Computer Science.
Since virtually all of my APC kids have had a strong Honors Physics
class I am able to cover material other than just mechanics. Prior to
the shift from 180x56 min to 90x90 min schedule I covered a more in
depth study of light to include as a last step measuring the Index of
Refraction of air using an interferometer. This led very nicely into
relativity. I then switched into electricity where I had them do a Wet
Chemistry experiment to measure the coulomb via the mole. We also did
some complex circuit analysis (multiple bridges and Kirchoff’s Laws)and
then went to magnetism where we built large coils of various sizes and
verified (sorry modelers) the laws derivable from Bio-Savart via direct
integration (yea, more Calculus). Then we actually measured the field
inside a ‘real’ solenoid via direct measurement then as a finale we
measured the mass of the electron via this solenoid. I then a quickie
experiment to determine Plank’s constant while electricity was fresh in
their minds. The last material was doing the traditional Bohr atom
deriviation. The nice thing is that we have already measured every
constant except for Coulomb’s Law which I have never been able to get
to work in the late Spring in NC. We also do the little Hydrogen lab
and compare results. That used to be my APC course. Alas, the
mechanics is the same but with 30+ hours gone I now have the kids build
optical slits then go straight to the interferometer in a quick demo
mode. I cover relativity in 2 days (ZOOM). I no longer do any
circuitry work nor measure the Coulomb. I jump entirely over BiotSavart and verification and we simply measure the magnetic field in a
solenoid then I give them the general equation and we compare. I still
do the mass of the electron simply because it is so cool. No longer do
Plank but I still wrap up with Bohr theory.
SOMEBODY PLEASE FIND THE IDIOT WHO CREATED BLOCK SCHEDULING!!
Warning about enrollments. My state (NC) requires Biology, Earth
Science and another ‘Lab Science’ of every graduate. Our AP
Environmental gets jazzed up to cover the state curriculum for Earth
Science. The 4 AP sciences get 9 sections, 1 Bio, 1 Chem, 1 Phys and 6
Envir. Needless to say, the state mandated Earth Science has really
wacked our AP Physics enrollment. We still have 5 sections of
Honors/Regular Physics so we are holding our own there. Most schools
in NC have suffered large drops in Physics (of any level) due to the
Earth Science requirement ours included as we dropped from 7/8 sections
down to 5 sections.
I have never taught the B course simply because the schools in NC that
my students attend typically ignore APB scores. I have considered
offering it as a first year course but I doubt if it would increase
enrollment since the hard core kids will be there anyway.
David Green
Teacher in Residence
Teacher of Physics
UNC-CH
Jordan HS
Phillips Hall, CB#3255
6806 Garrett Rd.
Chapel Hill, NC 27599-3255 Durham, NC 27707
(919) 962-5001
(919) 560-3912
greendl@physics.unc.edu
david.green@dpsnc.net
As always Gardner has some good information and advice here. As someone who teachers Mechanics C to
juniors predominantly in trig/pre-calc, I differ with some of his conclusions. The level of calculus required
to do well on the Mechanics C test is not at the AP Calculus BC or AB level. It can be taught to students
during the year with a total time of about 2 weeks. They need to know X^n derivatives and integrals. They
need to know how to express v and a in derivative (or 2nd derivative) form. They need to know how to
determine max and min of X^n functions. They need to have experience (preferably lab and computer
simulation) with velocity dependent forces and know how to qualitatively describe how velocity and
acceleration vary with time for objects in free-fall, moving horizontally, and up or down an incline. With
this much preparation your students can do well on FR and MC problems with calculus.
Occasionally other more involved calculus problems will come up. Some of these are challenging for
students who are concurrently in calculus. Some even challenge students who have completed a calculus
course. Some of these are beyond what is expected of students in their first semester of engineering level
physics. Take, for example, solving a differential equation. I took a look at 6 widely used college level
calculus physics books. Serway and Jewitt (both Physics for Scientists and Engineers and Principles of
Physics) mention differential equations in the velocity dependent force section early in the book but remark
that they are unfamiliar to the students at this point and do not expect students to solve them. They are not
mentioned again in these 2 texts. HR&W 8th mention them first on page 831. Three texts, Serway and
Vaughn, Knight, and Cummings don't mention them at all. Solving differential equations has come up 4
times in the entire history (back to the Summer of Love) of the Mech. C exam. 1984, 1993, 1999, and 2008.
It was worth 3 points in 1984, in 1993 it was worth 5 points. My students would have known how to get at
least 1 because they know how to write NSL as a differential equation (replace a with dv/dt). In 1999 it was
worth only 2 points! My students would have known how to get half of this simply by changing a to dv/dt.
I don't know how many points it was worth this year but I know it was probably not worth the time it would
take to do this at least 12-step process. In fact, I am pretty sure some of my calculus students will do worse
on this problem because they wasted time working on this difficult solution rather than just writing a =
dv/dt and getting on with it. Solving differential equations is beyond the scope of a first semester college
engineering physics class. The fact that the CB puts it on the test every 10 years or so should not prevent
anyone from considering teaching Mechanics C. I have found that my students appreciate calculus more
after seeing what it can be used for. Requiring calculus to be taken before taking Mechanics C would be
like teaching a wood shop class where all they ever do is study the schematics of the power tools. My
students may not learn how to use all the tools of calculus but at least they get to build something. I agree
that ideally they would be taking calculus concurrently. Making this a requirement at my school would
prevent many of my best students from taking AP Physics. Even if all my students were in calculus I would
not waste time solving diffy-Qs in my AP Mechanics C class.
I could go on about determining moments of inertia too. IMO it is of limited value to require students to do
this for thin rods only. This, too, is a more advanced topic best left for when they can also determine I of
regular solids too. I didn't solve for Is until taking an advanced Theoretical and Applied Physics class in
college as a sophomore. My students do just fine with the parallel axis theorem when asked to derive I of a
thin rod.
Dan Burns
Los Gatos High School
(where we expect to beat the national average on the 2008 Mech. 1 problem)
This important topic may be illustrated by the 2008 C Mechanics test. See,
for example, 2008M1 (the problem I was assigned at the Reading that just
ended).
1) Students need to understand that NSL is an equation with a derivative.
That is, after all, what "differential equation" means).
2) Then they need to solve it. The particular example, which requires
separation of variables, is about the hardest calculus required for any
recent AP exam.
3) Resistive velocity-dependent forces are often tested
The 2002 C Mechanics exam twice required integration of equations in the
form:
f(X) = A(B+CX)^-1
These examples show that students (even those concurrently in Calculus BC)
will need to be taught some calculus if you want to do the examples as they
come up in your curriculum. Granted, integration of these functions is
certainly done by the 2nd semester of Calculus BC (and maybe somewhat
earlier, if you have influence with and work well with the math department
at your school), but dealing with resistive forces occurs in the natural
sequence of most Physics C courses much earlier than that.
However, that is always the case with physics and math. The Physics C course
is designed to be taught with students concurrently in calculus (preferably
on the level of BC Calculus).
People who try to teach the C Mechanics course without calculus are probably
doing their students a disservice. The Development Committee has, as one of
its checks now, that use of Calculus is required somewhere. Years like 2003
(which required no calculus at all on the FR) probably won't happen again.
As I mentioned before, I was assigned to the 2008M1 problem this year. It
was very sad to see the many students (who clearly new some physics, judging
from the rest of their answers) not able to even attempt the calculus
portion.
Providing a place for these students is one of the motivators for the B
redesign.
Gardner Friedlander
I do Mech. C only. I think you will have to go into more detail in E&M
but
not much. I will use the terms "dot product" and "scalar product" and
point
out resources in their physics book for additional information. I don't
emphasize the rote mathematical formula, abcos(phi). Instead I talk
about
determining the component of the force in the direction of motion to
calculate the work done. I show how a force applied perpendicular to
the
direction of motion can't change the kinetic energy so no work can be
done
by it. I don't say that the dot product is always a cosine. We show how
to
determine the component of the force in the direction of motion using
whatever angles are given in the problem. I intentionally give them
problems
where the angle is given so that a sine is used.
I don't go into details about the cross product either. The full
treatment
is only needed for 3D problems which are not treated in my class. We
show
that torque is only produced by the component of force perpendicular to
the
lever arm. This is easy to demonstrate by poking the door handle in a
direction parallel to it. We say that torque is the component of force
perpendicular to the lever arm times the lever arm. I also show the
other
way to think about it, torque is the force times the component of the
lever
arm perpendicular to the force. I don't emphasize absin(phi) or do any
matrices. I give them problems here the angle is given so that a cosine
might be used.
I treat the angular momentum of a particle in a similar way. A particle
only
has angular momentum if it has a component of velocity perpendicular to
the
lever arm. This can be demonstrated by shooting a projectile parallel
to a
rotating platform. When it hits, the platform will not rotate. Angular
momentum is the only place where we talk about the direction of a
vector
that is the cross product of 2 other vectors. I sit on a chair that can
spin
freely with a spinning bicycle wheel with the plane of rotation
parallel to
the ground. When I flip it, L has to keep the same direction. That
means I
will have to start spinning in the direction the wheel was spinning
before I
flipped it. It helps to have a wheel with a large I for this demo. We
use
the right hand rule to predict which way I will spin given the wheel's
direction and to determine which way the wheel was spinning after
seeing
which way I spun. We also use the RHR to determine the direction of L
for
various spinning systems diagrammed on the board.
Dan Burns
Los Gatos High School
(where students need a pencil, a calculator, and a right hand for the
rotation test)
Below are my answers and comments to your questions. There are a lot of factors that go into choosing
what will work best at your school. I have primarily juniors in trig/pre-calc that have not had a physics
class. I have 5, 50-minute periods a week for about 31 weeks before the AP test. I realize that other schools
have different situations and make a different decision about what works best.
Dan Burns
Los Gatos High School
(where calculus in physics is at Mechanics C level)
-----Original Message----From: Paul Lulai [mailto:plulai@stanthony.k12.mn.us]
Sent: Fri 6/27/2008 1:36 PM
To: AP Physics
Subject: RE: [ap-physics] Calculus level of AP C
A question to Dan and others that teach AP C without calc as a PRE-requisite (sorry to shout).
Do you do anything to make sure kids are not scared of the class simply by the title? I am in a small school
(graduate ~140 each year). We have btn 20 - 50 kids each year take appb. About ½ of those are in
calculus. I think the pace of APC Mech would help my kids do even better (assuming the same crop of
kids). I see a few problems and questions.
1-
Kids will get scared of the title (more than already) AP Physics C Mechanics
>The students have never paid that much attention to the official name of the class. It was just called AP
Physics for the first 5-6 years I taught it. We then changed the name to AP Mechanics C and I didn't here
one comment from any of my students. I still need to stop by the guidance office to make sure they signed
up for the right test. I have had some parents ask if we are "only doing mechanics". I respond that we are
learning in depth the 3 great conservation principles and Newtonian mechanics which can be used to
understand a diverse amount of physics applications.
2-
My crop of kids changes due to the title. If I drop in numbers, I'm shot.
When I switched from AP B to AP Mechanics C, my enrollment grew. I started out my first year with 35
students, I have 118 enrolled for next year. IMO, more students can handle the material if given more time
and you spiral back through some of the topics.
3If I get ready to change for 09-10 (too late for 08-09), am I missing the boat on how the changes to
appb & appc will help our kids?
My understanding is that AP C will not change. I expect more schools to consider AP Mechanics C after
the change to B occurs. There are too many schools like mine that don't have room for 2 years of physics. I
also question whether colleges will value a score on only the first part of the AP B 2-year sequence. If you
will be unable to prepare them for both AP B tests, you should consider AP Mechanics C.
4I like doing more than just mechanics. Do you folks that teach JUST appc-mechanics also add some
other material or is that too much for kids learning calc on the fly?
I do Special Relativity in the 4-5 weeks after the AP Test. I also have done chaos theory and quantum
physics instead. If your students have had a prior physics class you could do a lot more. One logical topic
would be sound and light after the SHM unit. I have found that I need all of my time from August to May
to prepare them for the AP Mechanics C test. I also love teaching mechanics. I am always coming up with
new labs, activities, and approaches. If teaching just mechanics would affect your job satisfaction, the kids
would pick up on that so I wouldn't recommend doing it then.
5Do some teach appc-mechanics without any pre-requisite or co-requisiste of Calculus? I wonder
what a college thinks of a kid that has not taken calculus, yet earns a 3,4,or 5 on a calc based physics exam.
To my knowledge, no college has made this an issue for any of my students. They are more likely to get
credit for AP C Mechanics than for AP B if they are an engineering, physics, or architecture major. Since
most of my students are juniors, they take calculus as a senior anyway. The level of calculus needed for
Mechanics C is something you can easily teach to them in your physics class. My students learn this much
in their trig/pre-calc class. Unfortunately that is the unit they teach AFTER the AP exam.
Thanks for any insight.
In a message dated 6/30/2008 1:17:20 PM Eastern Daylight Time, David.Green@dpsnc.net
writes:
1. Actually derive using “wave the hand limits” the polynomial
derivatives.
2. Derive Sum, Difference, Product, Quotient and Chain rules.
3. Derive the slope of Sin and Cos and nicely get to show that
sinx/x = 1 and cosx/x = 0 . I also make a point of sinx<x<tanx
all approach zero at essentially the same rate. Trig functions
are useful once in awhile, mainly to “prove” that Ac = V^2/R
without using the rotating vector approximation and since I also
want to teach magnetism it is useful in direct integration of
some of the harder integrals(See way below for a caveat).
4. I also teach e^x and lnx and logx derivatives and integrals. In
fact, these are the ones that APC seems to use the most. As we
have seen several times in the past several years including the
strange year when it was on the test twice.
5. I also do a not so small lab where they deal graphically and via
Calculus with some data to see that it actually does work.
Wow, that's a great summary of the math background you give your APC kids. Are they taking
the Mech or E&M exam or both? Don't they need vector calculus for E&M? If so, what do they
need?
TIA,
AJG
A. Jorge Garcia
This is a great list of things that could be done using calculus in a high school physics class. I am saving it
for future reference and encourage other AP teachers to do so as well. I would like to point out that David
is preparing his students to score 90/90 on the Mechanics C test by doing all of this. He might be scaring
away some teachers from considering teaching AP Mechanics C with this extensive list. You can prepare
your students to do very well on the AP Mechanics C test by doing about half of this calculus. Since 55/90
is about where the 5 cutoff is, they don't need to score a 90. If you are inclined to do more like David, that
is great and your students will benefit from it.
Dan Burns
Los Gatos High School
(where we have never had someone score a 6 on the AP Mechanics C test)
In a message dated 7/1/2008 4:03:53 PM Eastern Daylight Time, DBurns@lgsuhsd.org writes:
I also make a point of sinx<x<tanx all
approach zero at essentially the same rate.
These all approach zero as x approaches zero and their drivatives are the same at zero, but
nowhere else unless you mean as an approximation near x=0?
HTH,
AJG
In a message dated 7/1/2008 4:03:53 PM Eastern Daylight Time, DBurns@lgsuhsd.org writes:
3.
Derive the slope of Sin and Cos and nicely get to show that
sinx/x = 1 and cosx/x = 0
I presume you mean these as limits when x -> 0?
Regards,
AJG
In a message dated 7/1/2008 4:03:53 PM Eastern Daylight Time, DBurns@lgsuhsd.org writes:
Sometimes I encourge my stronger students
to integrate the Fa=Kv^2 which they have found in doing the coffee
filter drop lab to determine predicted velocity and position.
Yes, I highlight solutions to diffeential equations too.
HTH,
AJG
On Jul 2, 2008, at 9:03 AM, David Green wrote:
AJG notes a small error in my methods:
“…
3.
Derive the slope of Sin and Cos and nicely get to show that
sinx/x = 1 and cosx/x = 0
I presume you mean these as limits when x -> 0?
…”
Yep!
Small correction: (1 - cos x)/x -> 0 as x -> 0. The ratio (cos x)/x is not defined as x -> 0.
David Derbes
U of Chicago Lab Schools
There is little need for the full three dimensional vector analysis.
Helpful with angular momentum, torque, and magnetism. Generally all
interesting mechanics problems at this level can be reduced to motion
in a plane. It's also entirely possible to do angular momentum, torque
and magnetism without the Full Monty of cross-products, using the
right hand rule and the law that the size of a cross product A x B is
AB sin theta, where theta is the angle between A and B, and the
direction is perpendicular to the plane containing A and B (either up
or down, direction given by right hand rule.)
I have only used the unit vectors i, j, k to derive the right hand
rule (showing kids the cross product as a determinant.) They're not
all that useful in my opinion; coordinates do as well.
Good luck!
David Derbes
U of Chicago Lab Schools