Transcript Physics 218: Lecture 2 Dr. David Toback Physics 218, Lecture II

Physics 218:
Lecture 2
Dr. David Toback
Physics 218, Lecture II
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Today’s Lecture
• Quick Overview of Chapter 1
– Significant Figures
– Units
– Diagrams
• Calculus 1
–
–
–
–
–
Change as a function of time
Velocity example
Limits
Derivatives
Acceleration example
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Purpose
• I want for you to do well in this class.
• I want to make it possible for everyone to get a
good grade and be a master problem solver.
My job is to make it clear what are the
important problems and teach you
straightforward ways to get good at solving
them.
• So: The purpose of this lecture is really to give
you a bunch of tricks so you can do well on the
problems in this class and later.
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Chapter 1: Introduction
This chapter is fairly well written. I won’t
lecture on most of it except for the parts
which I think are useful in helping you be a
better problem solver in general or for
helping you look like a professional.
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Number of Significant Figures
15 ± 1 feet (1 digit in uncertainty, same
“10’s” as last digit)
• 15.052 ± 1 feet (Makes you look like an
amateur)
• 15 ± 1.05 feet (Same thing)
• 15.1 ± 0.1 feet (Ok)
• 15 ± 10 feet (Ok)
An aside: Personally, I take significant digits
seriously. It makes you look bad when you mess
them up.
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Significant Figures
• Good test: Write the primary number as
1.5x101 feet (get rid of zeros on either end)
which is the “powers of 10 notation” or
what we call “scientific notation”
– 17526.423 = 1.7526423 x 104
• Then deal with the uncertainty
• Try one digit in the uncertainty
– Example: Fix 15.052 ± 1 feet
 (1.5052 ± 0.1) x 101 feet
 (1.5 ± 0.1) x 101 feet
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Units
• 10 feet is very different from 10 miles
• Area (size of a TV screen) can be given in
square inches in2
• Paying attention to the units will help you
catch LOTS of mistakes on exams,
quizzes and homework!!
– If we ask for the size of a cube, make sure your
answer in in feet
• Units MUST always be given. (Especially on
exams!) Wouldn’t want to look bad would you?
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Converting Units
• Multiplying anything by 1 (no units!) is a
GREAT trick! Use it often!!
– 1 meter x 1 = 1 meter
– 1 yard x 1 = 1 yard x (3 feet/yard) = 3 feet
(simple! Units cancel out!)
– Example:1 football field in feet
• 1 football field x (1) x (1) = 1 football field
• 1 football field x (100 yards/1 football field) x
(3 feet/yard) = 300 feet
• Both are units of length!
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Example Problem
You want to measure the
height of a building. You
stand 2m (2 strides) away
from a 3m pole and see that
it’s “in line” with the top of
the building. You pace off
about 16 more strides from
the pole to the building. What
is the height of the building?
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Problem Solving
• This class is mostly problem solving
(well… you need to understand the
concepts first in order to solve the
problems, but we’ll do both).
• In order to solve almost any problem you
need a model
• Physicists/engineers are famous for
coming up with silly models for
complicated problems
• The first step is always “Draw a
diagram!”
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Draw a Diagram!!!!
(Students who don’t draw diagrams don’t do
well…)
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Calculus 1
• Why are we doing math in a Physics class?
• Believe it or not, Calculus and Classical
mechanics were developed around the same
time, and they essentially enabled each
other.
• Calculus basically IS classical mechanics
• Bottom line: If you can’t do Calculus you
can’t REALLY do physics.
– It’s true you can do some simple problems
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Advice
• You really need to be comfortable
differentiating!
• If you aren’t, do lots of problems in a
introductory calculus book and take lots of
math quizzes.
• The “rate” at which things “change” will be
really big in everything we do.
• If you are struggling with the problems in
the handout get help now.
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Overview
• I’m not going to teach you calculus.
• The goals are:
– Teach (hopefully remind) you about how to think
about how things “change as a function of time.”
– Teach you how to take a derivative (and why you take
derivatives) so you can get by until you get to it in
calc.
• Diagrams are vital again!
• Units here will really help (there is a good
example of this in problem 1-9).
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Some Notation
• Let’s do some definitions
• Define “define”
– Example: t0  0 sec
– We can always make a definition, the idea is to
make one that is “useful”
– Another example: X = 22 feet  X0
• Define D as “the change in”
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Motion in One Dimension
• Car is moving
– X=0 feet at t0=0 sec
– X=22 feet at t1=1 sec
– X=44 feet at t2=2 sec
• We say this car has
“velocity”
• How do we get the
velocity from the
graph?
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Motion in One Dimension Cont…
•
•
Velocity: “Change in
position during a certain
amount of time”
Calculate from the Slope:
The “Change in position
as a function of time”
–
–
•
•
Change in Vertical
Change in Horizontal
Change: D
Velocity  DX/Dt
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Constant Velocity
Average Velocity  Velocity  v 
ΔX
Δt

X 2  X1
t 2  t1
This example:
X = bt
• Slope is constant
• Velocity is constant
– Easy to calculate
– Same everywhere
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Moving Car
A harder example:
X = bt2
What’s the velocity
at t=1sec?
Want to calculate
the “Slope” here.
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Derivatives
• To find the slope at time t, just take the
“derivative”
• For X=bt2 Slope = V =dx/dt =2bt
• “Gerbil” derivative method
–If X=
n
n-1
at V=dx/dt=nat
– “Derivative of X with respect to t”
• More examples
– X= bt2 V=dx/dt=2bt
– X= ct3 V=dx/dt=3ct2 (as in text)
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Common Mistakes
The trick is to remember what you are taking
the derivative “with respect to”
More Examples:
• What if X= 2a3tn?
– Why not dx/dt = 3(2a2tn)?
– Why not dx/dt = 3n(2a2tn-1)?
• What if X= 2a3?
– What is dx/dt?
– There are no t’s!!! dx/dt = 0!!!
– If X=22 feet, what is the velocity? =0!!!
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Check: Constant Position
– X = C = 22 feet
– V = dx/dt = 0
• Check
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Check: Constant Velocity
• Car is moving
– X=0 feet at t0=0 sec
– X=22 feet at t1=1 sec
– X=44 feet at t2=2 sec
• What is the equation of
motion?
• X = bt with b=22 ft/sec
• V = dX/dt
V= b = 22 ft/sec
• Check
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Check: Non-Constant Velocity
• X = bt2 with b=11 ft/sec2
• V = dX/dt = 2bt
• Car has “non-Constant”
velocity
• It is a “function of time”
• it “Changes with time”
– V=0 ft/s at t0=0 sec
– V=22 ft/s at t1=1 sec
– V=44 ft/s at t2=2 sec
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Acceleration
• We say that things which have changing
velocity are “accelerating”
• Acceleration is the “Rate of change of
velocity”
• You hit the accelerator in your car to speed
up
– It’s true you also hit it to stay at constant
velocity, but that’s because friction is slowing
you down
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Acceleration
• Acceleration is the “Rate of change of
velocity”
• Said differently: How fast is the Velocity
changing?
Accel 
ΔV
Δt

V2  V1
t 2  t1

dV
dt
• We’ll come back to this more next week
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More Derivatives
• X=bt2 V=2bt A= ?
• What about taking the
derivative of a
derivative?
• Rate of change of
Velocity
d dX
 
What about dV

dt
dt dt
Write this as
d 2X
dt 2
Derivation :
n
X  at
n-1
dX
 dt  nat
d2X
d dX



2
dt
dt
dt
n-1
d
 dt nat 
(n-1 )-1
 (n-1 )(na)t
n- 2
 (n-1 )nat
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Acceleration Continued
An example: (a, b and c are constants)
– X = a + bt + ct2
 V = dx/dt = 0 + b + 2ct
• Remember that the derivative of a
constant is Zero!!
– Accel = d2x/dt2 = 0 + 0 + 2c
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Position, Velocity and Acceleration
• Position, Velocity and Acceleration are all
related
– Velocity is the derivative of position with respect to
time
– Acceleration is the derivative of velocity with
respect to time
– Acceleration is the second derivative of position
with respect to time
• Calculus is REALLY important
• Derivatives are something we’ll come back to
over and over again
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X, V and A Cont…
• Calculus is REALLY important
• Derivatives are something we’ll come
back to over and over again
• The slope or the “rate” is often used
• Remember we want to know what
happens when we do an experiment
– What will happen as a function of time I.e.,
as the experiment progresses?
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Results of Math Evaluation
The average of all quizzes taken so far is about an
85 with a standard deviation of just above 10.
How to evaluate where you stand. If the average of
the scores of the quizzes you have taken is:
• 95 or above: Well prepared
• 85 - 90: Good, but needs to be better
• 80 - 85: Ok, but really needs some work
• 75 - 80: Hmmmm…maybe get some help
• 75 or below: Careful…Definitely get help! Maybe
drop…
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For Next Week
• Reading: Chapter 2
• Math Quizzes: Finish them!!
• On the schedule:
– Recitations meet
• Finish HW problems before section,
turn in after section
– Lecture: Chapter 2
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• Extra unused slides…
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Taking the Limit
V
X 2  X1
t 2  t1

X1 .5  X1
t1 .5  t1

X1 .1 X1
t1 .1 t1

X1 .0 0 1 X1
t1 .0 0 1 t1

(X1  DX )  X1
(t 1  Dt )  t1
• Limit as Dt0
–Gives the average
velocity between
t1 and t1+ Dt
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Easier Way
V
X 2  X1
t 2  t1
with t 2  t1 Dt and Dt small
Plug in with X  bt
2
X 2  b(t 2 )  b(t 1 Dt)
2
V

X 2  X1
t 2  t1

b(t1 Dt)2 - b(t1 ) 2
(t1 Dt) t1
b(t12 2 t1Dt  Dt 2 - t 12 )
Dt

2

b(t12 2 t1Dt  Dt 2 ) - bt12
Dt
b( 2 t1Dt  Dt 2 )
Dt

b( 2 t1  Dt ) Dt
Dt
 b(2t1  Dt) Limit

 2bt1
Dt 0
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Velocity
• Velocity is the slope
• We want the slope
at any given point
• Let’s start the
process and find the
slope between
points 1 and 2
• Clearly the drawn
line isn’t a great
approximation
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Slope
Average Velocity  Velocity  v 
• Successive approximations
Physics 218, Lecture II
ΔX
Δt

X 2  X1
t 2  t1
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• MLK day on Monday. No recitation on
Monday.
• Go to another section: Sections from my
class are best
– Wed, 10:20AM room 205
– Wed, 3:00PM, room 205
– If not, find one on the schedule
• There will be no quiz
• Don’t forget to get your lab manual
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