1. Physics 218, Lecture II1 Dr. David Toback Physics 218 Lecture 2

2. Physics 218, Lecture II2 In Class Quiz Write down the most important “student case study” from the Frequently Asked Questions handout

3. Physics 218, Lecture II3 Announcements: WebCT Having trouble getting started? Try: –ITS Help sessions –Open access lab/student computing –Instructions on faculty.physics.tamu.edu/toback/WebCT –email to webct@physics.tamu.edu Check your neo email account for announcements Still working on Math Quiz figures… sorry about that.. Finish your “Preliminary Course Materials”

4. Physics 218, Lecture II4 Due dates coming up Week 1 (This week): –Lecture: Chapter 1 (Reading, but nothing due) –Recitation & Lab: Lab 1 (A&B) –Homework due: None Week 2 (Next week): –Homework (Monday): Math quizzes –Lecture: Chapter 2 –Recitation & Lab: Chapter 1 and Lab 2 Week 3 (The week after that): –Homework due (Monday): Chapter 1 –Lecture: Chapter 3 & 4 –Recitation: Chapter 2 and Lab 3 Etc..

5. Physics 218, Lecture II5

6. 6 Chapter 1: Calculus Won’t cover the chapter in detail This is a chapter that is best learned by DOING We’ll cover it quickly –Lots more examples in Chapter 2 –Lots of practice in Math Quizzes on WebCT (when they’re fixed)

7. Physics 218, Lecture II7 Where are we going? We want Equations that describe Where am I as a function of time? How fast am I moving as a function of time? What direction am I moving as a function of time? Is my speed changing? Etc.

8. Physics 218, Lecture II8 Use calculus to solve problems!

9. Physics 218, Lecture II9 Motion in One Dimension Where is the car? –X=0 feet at t 0 =0 sec –X=22 feet at t 1 =1 sec –X=44 feet at t 2 =2 sec Since the car’s position is changing (i.e., moving) we say this car has “speed” or “velocity” Plot position vs. time –How do we get the speed from the graph?

10. Physics 218, Lecture II10 Speed Questions: How fast is my position changing? What would my speedometer read?

11. Physics 218, Lecture II11 How do we Calculate the speed? Define speed: “Change in position during a certain amount of time” Math: Calculate from the Slope: The “Change in position as a function of time” –Change in Vertical divided by the Change in Horizontal –Speed =  X  t Change: 

12. Physics 218, Lecture II12 Constant Speed Equation of Motion for this example is a straight line Write this as: X = bt Slope is constant Velocity is constant –Easy to calculate –Same everywhere Position time

13. Physics 218, Lecture II13 Moving Car A harder example: X = ct 2 What’s the speed at t=1 sec? Want to calculate the “Slope” here What would the speedometer say?

14. Physics 218, Lecture II14 Derivatives To find the slope at time t, just take the “derivative” For X=ct 2, Slope = V =dx/dt =2ct “Gerbil” derivative method –If X= at n  V=dx/dt=nat n-1 –“Derivative of X with respect to t” More examples –X= qt 2  V=dx/dt=2qt –X= ht 3  V=dx/dt=3ht 2

15. Physics 218, Lecture II15 Common Mistakes The trick is to remember what you are taking the derivative “with respect to” More Examples (with a=constant): What if X= 2a 3 t n ? –Why not dx/dt = 3(2a 2 t n )? –Why not dx/dt = 3n(2a 2 t n-1 )? What if X= 2a 3 ? –What is dx/dt? –There are no t’s!!! dx/dt = 0!!! –If X=22 feet, what is the velocity? =0!!!

16. Physics 218, Lecture II16 Going the other way: Integrals What if you know how fast you’ve been going and how long you’ve been driving How can you figure out how far you’ve gone? What would your car’s odometer read?

17. Physics 218, Lecture II17 Getting the Displacement from Velocity If you are given the speed vs. time graph you can find the total distance traveled from the area under the curve:  X=V 0 t + ½at 2 Can also find this from integrating… Slope is constant = Constant acceleration

18. Physics 218, Lecture II18 Definite and Indefinite Integrals

19. Physics 218, Lecture II19 Some Integrals

20. Physics 218, Lecture II20 Our Example

21. Physics 218, Lecture II21 For Next Week Before Lecture: –Read Chapter 2 –Math Quizzes due Monday In Lecture –Cover Chapter 2 Recitation, Lab and Homework: –Start Chapter 1 problems and exercises before recitation –Read your lab materials before lab

22. Physics 218, Lecture II22

23. Physics 218, Lecture II23 Simple Multiplication Multiplication of a vector by a scalar –Let’s say I travel 1 km east. What if I had gone 4 times as far in the same direction? → Just stretch it out, multiply the magnitudes Negatives: –Multiplying by a negative number turns the vector around

24. Physics 218, Lecture II24 Subtraction Subtraction is easy: It’s the same as addition but turning around one of the vectors. I.e., making a negative vector is the equivalent of making the head the tail and vice versa. Then add:

25. Physics 218, Lecture II25 Where am I? Traveling East then North is the same as traveling NorthEast Can think of this the other way: If I had gone NorthEast, the displacement is equivalent to having gone both North and East My single vector in some funny direction, can be thought of as two vectors in nice simple directions (like X and Y). This can make things much easier

26. Physics 218, Lecture II26 Problem Solving & Diagrams 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: Trick #2:“Draw a diagram!”

27. Physics 218, Lecture II27 Announcement: Free Tutoring Four foreign graduate students are available to tutor Physics 218 Students without charge. Students desiring help are to e-mail the tutor and arrange a time to meet in Heldenfels 211 on weekdays. The tutors are: Sunnam Min, smin@physics.tamu.edu Xi Wang, xwang@phyiscs.tamu.edu Rongguang Xu, rxu@physics.tamu.edu Hong Lu, hlu@physics.tamu.edu

28. Physics 218, Lecture II28 Components Let’s do this with the math: Break a vector into x and y components (I.e., a right triangle) THEN add them This is the sine and cosine game Can use the Pythagorean Theorem A 2 + B 2 = C 2

29. Physics 218, Lecture II29 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 at least helping you look like a professional

30. Physics 218, Lecture II30 Models, theories and Laws Prescriptive vs. Descriptive What should happen vs. What does happen when you do an experiment –US law doesn’t allow killing –Physics law shows clearly that it does happen.

31. Physics 218, Lecture II31 Estimating Order of Magnitude This is a useful thing to be able to do at home Let’s say you are at a grocery store and it’s full. How much will it cost you to buy it all? –Estimate using round numbers –50 items (assuming not lots of little things) –A dollar an item  $50

32. Physics 218, Lecture II32 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. Also, WebCT will do unpredectible things if you don’t use them correctly.

33. Physics 218, Lecture II33 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!

34. Physics 218, Lecture II34 Significant Figures Good test: Write the primary number as 1.5x10 1 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 10 4 Then deal with the uncertainty Usually only one digit in the uncertainty –Example: Fix 15.052 ± 1 feet → (1.5052 ± 0.1) x 10 1 feet → (1.5 ± 0.1) x 10 1 feet

35. Physics 218, Lecture II35 Reference Frames Frame of reference: Need to refer to some place as the origin Draw a coordinate axis –We define everything from here –Always draw a diagram!!!

36. Physics 218, Lecture II36 Vector notation: –In the book, variables which are vectors are in bold –On the overheads, I’ll use an arrow over it Vectors are REALLY important Kinda like calculus: These are the tools! First the Math: Vector Notation Some motion represented by vectors. What do these vectors represent physically?

37. Physics 218, Lecture II37 Adding vectors in funny directions Let’s say I walk in some random direction, then in another different direction. How do I find my total displacement? We can draw it It would be good to have a better way…

38. Physics 218, Lecture II38 Example We have two known displacements D 1 and D 2. What is the magnitude and angle of the net displacement in this example?

39. Physics 218, Lecture II39 Go home with a friend You are going home with a friend. You live in Houston and your friend lives in San Antonio. First you drive 100 miles SouthEast (known angle  ) from Aggieland to Houston, then 300 miles West to San Antonio? Using unit vector notation, what is your displacement from the center of the universe?

40. Physics 218, Lecture II40 Examples without an axis

41. Physics 218, Lecture II41 Addition using Components To add two vectors, break both up into their X and Y components… First break each vector into its X and Y components

42. Physics 218, Lecture II42 Addition using Components cont… Next: add separately in the X and Y directions Magnitudes of V F

43. Physics 218, Lecture II43 Drawing the components

44. Physics 218, Lecture II44 Vector Cross Product Cont… Calculating the cross product is the same as taking the determinant of a Matrix