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

2. Today’s Lecture Quick Overview of Chapter 1 Calculus 1
Significant Figures
Units
Diagrams
Calculus 1
Change as a function of time
Velocity example
Limits
Derivatives
Acceleration example
Physics 218, Lecture II

3. 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.
Physics 218, Lecture II

4. 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.
Physics 218, Lecture II

5. Number of Significant Figures
15 ± 1 feet (1 digit in uncertainty, same “10’s” as last digit)
± 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.
Physics 218, Lecture II

6. 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”
= x 104
Then deal with the uncertainty
Try one digit in the uncertainty
Example: Fix ± 1 feet
 ( ± 0.1) x 101 feet
 (1.5 ± 0.1) x 101 feet
Physics 218, Lecture II

7. 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?
Physics 218, Lecture II

8. Converting Units 1 meter x 1 = 1 meter
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!
Physics 218, Lecture II

9. 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?
Physics 218, Lecture II

10. 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!”
Physics 218, Lecture II

11. Draw a Diagram!!!! (Students who don’t draw diagrams don’t do well…)
Physics 218, Lecture II

12. 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
Physics 218, Lecture II

13. 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.
Physics 218, Lecture II

14. 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).
Physics 218, Lecture II

15. 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”
Physics 218, Lecture II

16. 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?
Physics 218, Lecture II

17. 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
Physics 218, Lecture II

18. Constant Velocity This example: X = bt Slope is constant
Velocity is constant
Easy to calculate
Same everywhere
Physics 218, Lecture II

19. Moving Car A harder example: X = bt2 What’s the velocity at t=1sec?
Want to calculate the “Slope” here.
Physics 218, Lecture II

20. Derivatives If X= atn V=dx/dt=natn-1
To find the slope at time t, just take the “derivative”
For X=bt2 Slope = V =dx/dt =2bt
“Gerbil” derivative method
If X= atn V=dx/dt=natn-1
“Derivative of X with respect to t”
More examples
X= bt2 V=dx/dt=2bt
X= ct3 V=dx/dt=3ct2 (as in text)
Physics 218, Lecture II

21. 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!!!
Physics 218, Lecture II

22. Check: Constant Position
X = C = 22 feet
V = dx/dt = 0
Check
Physics 218, Lecture II

23. 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
Physics 218, Lecture II

24. 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
Physics 218, Lecture II

25. 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
Physics 218, Lecture II

26. Acceleration Acceleration is the “Rate of change of velocity”
Said differently: How fast is the Velocity changing?
We’ll come back to this more next week
Physics 218, Lecture II

27. More Derivatives X=bt2 V=2bt A= ?
What about taking the derivative of a derivative?
Rate of change of Velocity
Physics 218, Lecture II

28. 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 = c
Physics 218, Lecture II

29. 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
Physics 218, Lecture II

30. 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?
Physics 218, Lecture II

31. 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
: Good, but needs to be better
: Ok, but really needs some work
: Hmmmm…maybe get some help
75 or below: Careful…Definitely get help! Maybe drop…
Physics 218, Lecture II

32. 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
Physics 218, Lecture II

33. Extra unused slides…
Physics 218, Lecture II

34. Limit as Dt0 Gives the average velocity between t1 and t1+ Dt
Taking the Limit
Limit as Dt0
Gives the average velocity between t1 and t1+ Dt
Physics 218, Lecture II

35. Easier Way
Physics 218, Lecture II

36. 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
Physics 218, Lecture II

37. Slope
Successive approximations
Physics 218, Lecture II

38. 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
Physics 218, Lecture II