Physics 218, Lecture III1 Physics 218 Lecture 3 Dr. David Toback

Physics 218, Lecture III2 Checklist for Today Things that were due last Thursday: –Chapter 1 reading –Read all handouts from web page Things that are due yesterday (Monday): –WebCT warm-ups (FCI, Math Assess, etc…) –Math Quizzes 1 through 10 Things that are due today: –Reading for Chapter 2 –Chapter 2 Lecture Questions For this week and/or due next Monday: –Recitation: Lab materials, start Ch. 1 on WebCT –All HW1 problems on WebCT due Monday

Physics 218, Lecture III3 Describing Motion Interested in two key ideas: How objects move as a function of time –Kinematics –Chapters 2 and 3 Why objects move the way they do –Dynamics –Do this in Chapters 4 and 5

Physics 218, Lecture III4 Chapter 2: M otion in 1-Dimension Today: Velocity & Acceleration –Equations of Motion –Some calculus (derivatives) Thursday: –More calculus (integrals) –Problems

Physics 218, Lecture III5

6 Notes before we begin This chapter is a good example of a set of material that is best learned by doing examples We’ll do some examples today Lots more next time…

Physics 218, Lecture III7 Equations of Motion 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 its velocity changing? Etc.

Physics 218, Lecture III8 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 “velocity” or “speed” Plot position vs. time –How do we get the velocity from the graph?

Physics 218, Lecture III9 Velocity Questions: How fast is my position changing? What would my speedometer read? What is my instantaneous Velocity?

Physics 218, Lecture III10 How do we Calculate Velocity? Define Velocity: “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 –Velocity =  X  t Change: 

Physics 218, Lecture III11 Constant Velocity 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

Physics 218, Lecture III12 Moving Car A harder example: X = ct 2 What’s the velocity at t=1 sec? Want to calculate the “Slope” here

Physics 218, Lecture III13 Math Digression: 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

Physics 218, Lecture III14 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!!!

Physics 218, Lecture III15 Check: Constant Position –X = C = 22 feet –V = slope = dx/dt = 0 Check

Physics 218, Lecture III16 Check: Constant Velocity Car is moving –X=0 feet at t 0 =0 sec –X=22 feet at t 1 =1 sec –X=44 feet at t 2 =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 III17 Check: Non-Constant Velocity X = ct 2 with c=11 ft/sec 2 V = dX/dt = 2ct The velocity is: “non-Constant” a “function of time” “Changes with time” –V=0 ft/s at t 0 =0 sec –V=22 ft/s at t 1 =1 sec –V=44 ft/s at t 2 =2 sec

Physics 218, Lecture III18 Acceleration If your velocity is changing, you are “accelerating” –You hit the accelerator in your car to speed up at a stop light (Ok…It’s true you also hit it to stay at constant velocity, but that’s because friction is slowing you down…we’ll get to that later…) How quickly is the velocity changing? That’s our Acceleration

Physics 218, Lecture III19 Acceleration Acceleration is the “Rate of change of velocity” Said differently: “How fast is the Velocity changing?” “What is the change in velocity as a function of time?”

Physics 218, Lecture III20 Example You have an equation of motion where: X = X 0 + V 0 t + ½at 2 where X 0, V 0, and a are constants. What is the velocity and the acceleration?  V = dx/dt = 0 + V 0 + at Remember that the derivative of a constant is Zero!!  Accel = dV/dt =d 2 x/dt 2 = a

Physics 218, Lecture III21 Position, Velocity and Acceleration All three are 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 III22 Important Equations of Motion If the acceleration is constant Position, velocity and Acceleration are vectors. More on this in Chap 3

Physics 218, Lecture III23 Conceptual Example If the velocity of an object is zero, does it mean that the acceleration is zero? If the acceleration is zero, does that mean that the velocity is zero?

Physics 218, Lecture III24 Car Crash Test Design You are designing a crash test setup for a car maker. You can accelerate a car from rest with a constant acceleration of 1.00 m/s 2 so you can make the car crash into a wall. (This is the last time you will see numbers in a problem in lecture). 1.If the path is 200m long, what is the velocity of the car just before/as it hits the wall? 2.For the same acceleration, if you want the car to hit the wall with a speed of 30m/s (about 60 mi/hr), what minimum length must you have?

Physics 218, Lecture III25 Next Time Textbook Reading and Reading Questions: –None (Chap 3 assigned on Thursday) Homework: Math Quizzes and Chap 1 –Math Quizzes were due Monday –Work all Ch 1 probs before recitation –Start WebCT for Ch 1 before recitation –Chapter 1 HW due next Monday Recitation: Ask TA for help on hard HW problems Lab: Read VP Manual before lab Thursday: more example problems

Physics 218, Lecture III26

Physics 218, Lecture III27 Decelerating Car You are driving a car along a straight highway when you put on the brakes. The initial velocity is 15.0m/s to the right, and it takes 5.0s to slow the car down until it is moving at 5.0m/s to the right. What is the car’s average acceleration?

Physics 218, Lecture III28 Examples Can a car have uniform speed and non-constant velocity? Can an object have a positive average velocity over the last hour, and a negative instantaneous velocity?

Physics 218, Lecture III29 Constant Velocity This example: X = bt Slope is constant Velocity is constant –Easy to calculate –Same everywhere

Physics 218, Lecture III30 More Questions on the Car Crash What is the distance traveled? What is the total displacement? What is the average speed? Is the average speed the same as the average velocity? What is the instantaneous velocity at all times?

Physics 218, Lecture III31 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 III32 If the motion started here, call this x 0 Displacement Where are you? I.e, What is your displacement? Well…relative to where? Example: I’m 10 blocks north east of Kyle field What do we need to know? Where does the motion start? x 0 ? x 0 is relative to the origin x 0 meters from the origin When does the motion start? t 0 ?

Physics 218, Lecture III33 Vectors vs. Scalars Scalar Distance traveled is 100m Vector Displacement is 40m East Let’s say we traveled on a path like in the figure Distance traveled from the origin is a Scalar (like your car odometer). Displacement from the origin is a Vector –Has a distance and a direction from the origin Speed is a scalar Velocity is a vector –Negative distance? –Displacement?

Physics 218, Lecture III34 Another reason to care about vectors It turns out that nature has decided that the directions don’t really care about each other. Example: You have a position in X, Y and Z. If you have a non-zero velocity in only the Y direction, then only your Y position changes. The X and Z directions could care less. (I.e., they don’t change). Represent these ideas with Vectors

Physics 218, Lecture III35 Acceleration An object is accelerating if it’s “velocity is changing as a function of time” –Acceleration = dv/dt Acceleration and velocity can be pointing in different directions –How? What is the difference between average acceleration and instantaneous acceleration?

Physics 218, Lecture III36 If the motion started here, call this x 0 Displacement Where are you? I.e, What is your displacement? Well…relative to where? Example: I’m 10 blocks north east of Kyle field What do we need to know? Where does the motion start? x 0 ? x 0 is relative to the origin x 0 meters from the origin When does the motion start? t 0 ?

Physics 218, Lecture III37 Average Velocity Average speed Average velocity Total time = 10sec Avg Speed = 100m/10s = 10m/s Avg Velocity = (40m East)/10s = 4m/s East

Physics 218, Lecture III38 Instantaneous Velocity Average and Instantaneous Velocity Average is “over a period of time” –I.e., How many miles you traveled in a day Instantaneous is how fast are you going “right now” Car example: –Instantaneous is more like your speedometer. –Average is taking how far you traveled in the last hour and and dividing by an hour (includes the stops at the gas station)

Physics 218, Lecture III39 Instantaneous Cont… V=  x/  t (use total change in x, t: average) (instantaneous) Magnitude of instantaneous velocity is always the same as the instantaneous speed –Why? In the last example, is the average velocity the same as the average speed? Distance and displacement become identical in the limit that they become infinitesimally small

Physics 218, Lecture III40 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 III41 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 This stuff is going to go by quickly!

Physics 218, Lecture III42 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 your calculus class Diagrams are vital again! Units here will really help (there is a good example of this in problem 1-9 on the Calculus handout).

Physics 218, Lecture III43 Some Notation Let’s do some definitions Define “define” –Example: t 0  0 sec –We can always make a definition, the idea is to make one that is “useful” –Another example: X = 22 meters  X 0 Define  as “the change in”