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• Количество слайдов: 44 Physics 218 Lecture 2 Dr. David Toback Physics 218, Lecture II 1 In Class Quiz Write down the most important “student case study” from the Frequently Asked Questions handout Physics 218, Lecture II 2 Announcements: Web. CT • Having trouble getting started? Try: – ITS Help sessions – Open access lab/student computing – Instructions on faculty. physics. tamu. edu/toback/Web. CT – email to [email protected] tamu. edu • Check your neo email account for announcements • Still working on Math Quiz figures… sorry about that. . • Finish your “Preliminary Course Materials” Physics 218, Lecture II 3 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. . Physics 218, Lecture II 4 Physics 218, Lecture II 5 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 Web. CT (when they’re fixed) Physics 218, Lecture II 6 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. Physics 218, Lecture II 7 Use calculus to solve problems! Physics 218, Lecture II 8 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? Physics 218, Lecture II 9 Speed Questions: • How fast is my position changing? • What would my speedometer read? Physics 218, Lecture II 10 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: D – Change in Vertical divided by the Change in Horizontal – Speed = DX/Dt Physics 218, Lecture II 11 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 Physics 218, Lecture II time 12 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? Physics 218, Lecture II 13 Derivatives • To find the slope at time t, just take the “derivative” • For X=ct 2 , Slope = V =dx/dt =2 ct • “Gerbil” derivative method – If X= n at n-1 V=dx/dt=nat – “Derivative of X with respect to t” • More examples – X= qt 2 V=dx/dt=2 qt – X= ht 3 V=dx/dt=3 ht 2 Physics 218, Lecture II 14 Common Mistakes The trick is to remember what you are taking the derivative “with respect to” More Examples (with a=constant): • What if X= 2 a 3 tn? – Why not dx/dt = 3(2 a 2 tn)? – Why not dx/dt = 3 n(2 a 2 tn-1)? • What if X= 2 a 3? – 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 15 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? Physics 218, Lecture II 16 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: Slope is constant = Constant acceleration X=V 0 t + ½at 2 • Can also find this from integrating… Physics 218, Lecture II 17 Definite and Indefinite Integrals Physics 218, Lecture II 18 Some Integrals Physics 218, Lecture II 19 Our Example Physics 218, Lecture II 20 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 Physics 218, Lecture II 21 Physics 218, Lecture II 22 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 Physics 218, Lecture II 23 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: Physics 218, Lecture II 24 Where am I? Traveling East then North is the same as traveling North. East Can think of this My single vector in some funny direction, can be thought of as the other way: two vectors in nice simple If I had gone directions (like X and Y). This North. East, the can make things much easier displacement is 218, Lecture II Physics 25 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!” Physics 218, Lecture II 26 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, [email protected] tamu. edu • Xi Wang, [email protected] tamu. edu • Rongguang Xu, [email protected] tamu. edu Physics 218, Lecture II 27 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 Physics 218, Lecture II 28 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 Physics 218, Lecture II 29 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. Physics 218, Lecture II 30 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 Physics 218, Lecture II 31 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, Web. CT will do unpredectible things if you don’t Physics 218, 32 use them correctly. Lecture II 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) = 1 football field – 1 football field x (100 yards/1 Physics (3 feet/yard) = 300 33 football field) x 218, Lecture II Significant Figures • Good test: Write the primary number as 1. 5 x 101 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 • Usually 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 Physics 218, Lecture II 34 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!!! Physics 218, Lecture II 35 First the Math: Vector Notation • Vector notation: – In the book, variables which are vectors are in. Some motion represented by vectors. bold What do these vectors represent physically? – On the overheads, I’ll use an arrow over it • Vectors are REALLY important • Kinda like Physics 218, Lecture II calculus: These 36 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 Physics 218, Lecture II way… 37 Example We have two known displacements D 1 and D 2. What is the magnitude and angle of the net displacement in this example? Physics 218, Lecture II 38 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 South. East (known angle Q) 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? Physics 218, Lecture II 39 Examples without an axis Physics 218, Lecture II 40 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 Physics 218, Lecture II 41 Addition using Components cont… Next: add separately in the X and Y directions Magnitudes of VF Physics 218, Lecture II 42 Drawing the components Physics 218, Lecture II 43 Vector Cross Product Cont… Calculating the cross product is the same as taking the determinant of a Matrix Physics 218, Lecture II 44 