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

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

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

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

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. Physics 218, Lecture II

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 Physics 218, Lecture II

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Physics 218, Lecture II

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

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

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… Physics 218, Lecture II

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

Extra unused slides… Physics 218, Lecture II

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

Easier Way Physics 218, Lecture II

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

Slope Successive approximations Physics 218, Lecture II

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