LECTURE 1
1. Purpose of Physics 225
Bridge to fill in the gaps between PHYS/MATH 2xx and upper-level courses (PHYS 325 onwards)
Problem #1: PHYS 21X courses often have > 1,000 students, from many different curricula
 we can’t use the level of math that we would like to for physics majors’ intro courses
e.g. can’t use complex numbers, 3D differential calculus, serious integrals,
 some math / math application gets lost between MATH 2X1 and PHYS 325++
e.g. Taylor approximations, setup of multi-D integrals, use of complex numbers,
non-Cartesian unit vectors (not covered at all in MATH 241), …
 but upper-level courses assume you’ve had plenty of practice applying
calculus to elementary physics problems
Problem #2: Move Special Relativity out of PHYS 325 to make room for other material
 moved Lagrangian mechanics from 326  325 (as not everyone takes 326 anymore)
 added some fluid dynamics to 326 (not covered elsewhere)
Problem #3: Physics majors/minors can’t find each other in vast sea of 21X students!
Things to be aware of: using style of an upper-level course, which means
 you must TAKE NOTES YOURSELF (both discussion & lecture)
 problems are NOT SINGLE-STEP, and *never* multiple choice
 homework & exams are HANDWRITTEN: must show *reasoninig*
 averages on exams will be LOW: typically 60%  don’t freak! this is normal!
 GO TO OFFICE HOURS  you may not have needed to in 21X, but go now!
Sequence of material each week
1. learn it in Discussion: discover & derive new things yourself
2. wrestle with it in Homework: solve as much as you can.
3. refine it in Lecture: we'll tackle example problems similar to those on the homework to
illuminate key techniques, answer all your questions, and cover some subtleties
4. wrap it up in Office Hours: we will happily go through all of your homework solutions before
you submit them, to make sure the week's material is 100% clear. Homework is
*Training*, it is *not* a quiz, and the goal is to provide you with enough help to get
full marks every time.
The order of points 2 and 3 is key, and reflects my experience of how I learn things:
==> It is *USELESS* to listen to an expert (lecture instructor, conference speaker, posted solution, etc)
unless you have *ALREADY*wrestled with the material yourself.
Practical Matters
 Weekly Sequence for introduction of material
 Homework: due every Thursday 11 am in homework box; posted on web previous Friday
 Homework grading: must show reasoning! no work, no points
 Exams: 1 hour midterm in lecture period of week 8 worth 15%, 3 hour final worth 40%
 Other grades: homework 35%, discussion attendance 10%
 Office hours: Greg Monday 4 – 6 pm for 1 hour on Tuesdays after lecture;
TA’s on Sunday 3 – 5pm and Wednesday 5 – 7 pm
2. The Physicist’s Toolbox  Tools 1 & 2
Introduce quickly the 3 tools that experienced physicists apply to problems automatically:
 Units!
 Sketches with labels!
 Limiting cases!  coming up in a couple of weeks
2.1 Units
Imagine you are doing a real calculation in real life, for which there’s no answer at the back of any book
and on which actual money depends … you’ll find yourself checking units at every line of your work
without even thinking about it!
Example: You’re doing summer research in astrophysics and your adavisor asks you to calculate the
gravitational force between a meteor of mass m and a gas nebula of some complicated structure with
total mass M. The size and shape of the nebula are characterized by two length parameters d and D, and
the distance between the centers of the two objects is r.
Do you email these answers to your supervisor?
(a) F 



With a bit of practice looking for this mistake at every line, you’ll spot it without thinking.
Happens easily! e.g. mis-copying from one line to the next
Exam scoring: automatic –2 points for any final answer with mixed units (motivation! ;-))
(b) F 


GMm D
 MIXED UNITS: can’t add / subtract apples & oranges
r(r  d 2 )
GMm D
r(r  d)
 OVERALL UNITS
Easiest way to check: compare with any similar formula you can remember, e.g. F=GMm/r2
Exam scoring: automatic –1 for any final answer with incorrect units unless you notice it
and write that down. (“I know I made an error because …” is worth points!)
Example: Actual final exam question + solution from an actual student (scanned below)
A thin rod has length 2L, is placed along the z-axis, and carries a non-uniform charge per unit
length (z) = z where  is a constant with units C/m2. Calculate the electric field E(z) at points
on the z axis above the rod (i.e. for z > L).
(a) Can you spot the error in the final boxed answer?
Answer: Mixed units in parentheses! 4L/(z2–L2) has dimensions, log is dimensionless.
(b) Can you find where the error occurred? Debugging technique: search backwards until you find a
point where the units were ok.
Answer: End of 2nd-last line is ok  everything in parentheses is dimensionless like
the logs. The bug is in the last step where the student’s “z” looks so much like a “2”
that (z2 + zL – z2 + zL) became “4L” 
2.2 Sketches with Labels: survival tool for real-world multi-step problems
Go straight to the first problem. Give them 4 minutes to work on it, then ask for some
answers for the crowd, and ask by show of hands how many agree with each one … then solve the
problem in detail showing good technique clearly and making these 4 points as they come up:




Make! A! Sketch! Don’t just grab the first formula you can think of!
Invent symbols for every parameter  don’t work directly with numbers! Makes algebra
harder, unit checking impossible, and impossible to keep track of what you’re doing in multi-step
problems!
Indicate clearly what you know and what you want to know  essential for strategy!
Don’t be concerned if you have to introduce a parameter that is neither a given quantity nor the
thing you want to solve for. Student from previous class: “I always feel that when I have to
introduce a variable that wasn’t given, I’m making a mistake!”  this feeling is a disease caused
by solving too many one-step problems!!! It is essential to introduce intermediate parameters in
most real-world = multi-step problems!
Example: Two speed skaters, Alice and Bob, are at the end of a race. As Alice passes the last-lap
marker, she is traveling at a constant speed of 20 m/sec. Bob is behind: he passes the last-lap marker
5 sec after Alice does, but he is traveling at 60 m/sec. His faster speed is just enough to catch up to Alice
and the race ends in a tie. How far was the finish line from the last-lap marker?
Important solution techniques:

Sketch! Best one in this case is actually an (x,t) PLOT of the two trajectories, but
whatever lets you visualize what’s going on is fine!

Sketch with labels! Pick symbols for everything, make them meaningful, and do NOT
work with numbers directly. (Putting numbers in early makes algebra much harder and
completely prevents unit- and limiting-case checking.)

Strategy Box: Summarize these quantities in a box next to the figure:
What do we know and what do we want to know?
 Know: vA = 20 m/s, vB = 60 m/s, tB = 5 s
 Want: d = ?
I often circle the goal “d = ?” so I always know where I’m heading; otherwise you can
easily get lost in multi-step problems.

Introduce a parameter, in this case for t = time of race end  it’s normal to do so in
multi-step problems. We know from our strategy that we must get rid of it … easy, set
up 2 equations for t & equate them. For this problem, the equations are:
(a) t = d / vA
(b) t = tB + d / vB
From our strategy box, we want d, while we neither know nor want t. Thus we get rid of
t by combining the two equations (e.g. solve for t with one of them and substitute
into the other)  d / vA = tB + d / vB
Answer:
d
vA vB t B 20  60  5

 150 m
vB  v A
40
Example: A cannonball is launched at an initial speed v0 and at a launch angle of  with respect to the
ground. Calculate the launch angle  that maximizes the horizontal distance that the cannonball travels
before it hits the ground.
Similar technique points as the previous problem:

Sketch: symbols are given here but put them on the sketch anyway, as we must be able
to visualize the problem before we can solve it effectively

Introduce a parameter: total time T of cannonball trajectory

Strategy: what do we know?  v0, , constant g … what do we want to know?  x() = ?
Answer:
2v02
v02
x
sin  cos   sin 2
g
g
 max at  = 45