Physics 111. Help sessions meet Sunday, 6:30-7:30 pm in CLIR Wednesday, 8-9 pm in NSC 098/099

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1 ics Announcements day, ember 7, 2007 Ch 2: graphing - elocity s time graphs - acceleration s time graphs motion diagrams - acceleration Free Fall Kinematic Equations Structured Approach to Problem Soling Help sessions meet Sunday, 6:30-7:30 pm in CLIR Wednesday, 8-9 pm in NSC 098/099 Announcements Announcements Class time will be more aluable if you come prepared - read the book - work recommended problems - ask quesitons! You must bring both your Lab Manuals and Lab Notebook to Lab or you will not be allowed to begin this week. There will be another short quiz at the beginning of lab. Ball Motion Diagrams - Aerage Speed page 10 1) A, E, C, F, D, B 2) E, F, C, D, A=B 3) E, C, D, A=B=F 4) E, C, D, A=B, F 5) C, D, A=B, F, E 6) C, D, A=B=F, E 7) None of the Aboe What if the elocity of our object is NOT constant? How would we find the aerage elocity oer the time interal marked aboe? The aerage elocity is simply the slope of the green line!

2 Notice that an object moing with uniform elocity during the time period between the red lines has the same displacement as the object moing with the non-uniform elocity during the same time interal. t 1 t* t 2 Does the elocity of the object = the aerage elocity for the entire time interal [t 1, t 2 ]? Is its elocity less than, equal to, or greater than the aerage elocity during the time interal [t 1, t 2 ]? t 1 t* t 2 We call the object elocity,, at each instant in time the t 1 t* t 2 We call the object elocity,, at each instant in time the And we can find the instantaneous elocity of an object by finding the slope of the cure on the s. graph! For what length of time does the instantaneous elocity describe the elocity of an object in motion, such as the one depicted in the graph aboe? If we can describe the of the object with some function of time, say x(t), then the (instantaneous) elocity of the object is gien by r (t = t * xt*+ε x r ) = lim t* ε 0 (t * +ε) t * Out-of-gas Draw a motion diagram for the following problem. Make sure to include elocity ectors: A car rolls to a stop on a flat road after running out of gas. Class Worksheet #1 To more completely describe the motion of our the car, we d like to describe how its elocity is changing... We can use the motion diagram to help us t 4 t 3 t 2 t 1 5,4 4,3 f 3,2 2,1 1,0

3 The change in elocity is called acceleration. To find the acceleration, start by subtracting successie elocity ectors. r = r r 2,1 2,1 r 1,0 t 4 t 3 t 2 t 1,0 1 5,4 f 4,3 3,2 2,1 1,0 The change in elocity is called acceleration. To find the acceleration, start by subtracting successie elocity ectors. r 2,1 1,0 + r = r 2,1 t 4 t 3 t 2 t 1,0 1 5,4 4,3 3,2 2,1 1,0 f The change in elocity ectors are proportional to the acceleration ectors: 5,4 f a i = r i = i+1,i r i,i 1 t (t j +1 1 )/2 a 1 2,1 t 4 t 3 t 2 t 1,0 1 4,3 3,2 2,1 1,0 If the time interals are constant, this formulation reduces to: a i = r i = i+1,i r i,i 1 t t a 1 2,1 t 4 t 3 t 2 t 1,0 1 5,4 4,3 3,2 2,1 1,0 f Repeat the process to determine the acceleration ectors where possible. Note that without more information, we cannot find the acceleration at times and. a 4 a 3 a 2 a 1 The picture below represents a complete motion diagram. You can now use this tool to examine the motion in a ariety of situations. You ll find that motion diagrams help you to aoid many pitfalls a 4 a 3 a 2 a 1 5,4 f t 4 t 3 t 2 t 1 4,3 3,2 2,1 1,0 t 4 t 3 t 2 t 1 5,4 4,3 3,2 2,1 1,0 f

4 Class Worksheet #2-3 Bowling Ball Draw a motion diagram for the bowling ball. Experiment: Ball rolls up the incline. Experiment: Ball rolls up & down the incline. Let s now look at the graphs for an object which is not moing with uniform elocity. First, construct a graph of its elocity ersus using the information in the s. graph. Class Worksheet #4 s t elocity t i This graph proides a useful starting poinrom which we can determine how fast the elocity is changing. (clock reads ) r (clock reads t i ) elocity t i This graph proides a useful starting poinrom which we can determine how fast the elocity is changing. (clock reads ) r (clock reads t i ) Is this quantity greater than zero, less than zero or equal to zero for our moing object? Is the object speeding up, slowing down, or moing with uniform elocity? (clock reads ) (clock r reads t i ) A car accelerates at 2.5 mi/hr/s. Its instantaneous elocity is 20 mi/hr. What will be the elocity of the car at the end of the next second? And the one after that? Aerage acceleration oer the time interal [t i, ] Is usually written... a = r r t = f r i If the car is accelerates at 2.5 mi/hr/s, then at the end of the next second, the car will be traelling with the elocity = 20 mi/hr mi/hr = 22.5 mi/hr. At the end of the next second the car will be traelling with elocity = 22.5 mi/hr mi/hr = 25 mi/hr.

5 What is free fall? Is the acceleration uniform during free fall? Class Worksheet #5: Free Falling Ball For the ball falling from my hand, sketch s elocity s acceration s (coordinate system: x > 0 aboe the floor) Is there such thing as an instantaneous acceleration? If so, what does it mean? NOTE: Most of the problems in this class will inole objects moing with uniform or zero acceleration. (uniform acceleration ONLY) f = r i + a( t) r x f = x r i + r i ( t) + 1 r a( t) 2 2 a = constant where t These equations are easy to derie using calculus. How do we interpret these relationships graphically? (uniform acceleration ONLY) a = constant x f = x r i + r i ( t) + 1 r a( t) 2 f = r i + a( t) r 2 How do we interpret these relationships graphically? (uniform acceleration ONLY) a = constant x f = x r i + r i ( t) + 1 r a( t) 2 2 elocity The area under the cure is the distance traeled by the object oer the time interal. elocity Area of rectangle = i f f = i + a =a = f - i = a i Area of triangle = a2 /2

6 What if the acceleration is NOT constant? What if the acceleration is NOT constant? elocity The area under the cure is still the distance traeled by the object oer the time interal. elocity The area under the cure is still the distance traeled by the object oer the time interal. t x = (t) t Split up the interal of interest into a bunch of little time interals in which the elocity is approximately constant. dt d = x f x i = lim t 0 x = (t) t We now hae enough background to begin soling problems! This approach inoles 3 steps, two of which we already hae coered: Let s set up a procedure for problems soling that is thoughtful and organized. Our procedure needs to be ersitile enough to help us work through a wide range of problems of aried difficulty. 1) ical Representation 2) Pictorial Representation (new!) 1) ical Representation Motion Diagrams We ll learn other ical Representations in the near future Definitions of, displacement, elocity, and acceleration Kinematic equations More to come!

7 2) Pictorial Representation (New!) Draw pictures of the physical situation at all the key times in the problem Define the coordinate system Define the ariables List knowns and unknowns The approach is best demonstrated with a problem. Let s try this one together: George lies in Chicago and want to meet up with his friend Annabel for lunch, but Annabel lies in Pittsburg, 400 miles East of Chicago. If George leaes home at 9 am traeling 60 mph East and Susan leaes at the same time traeling 40 mph West, where will they meet? George lies in Chicago and want to meet up with his friend Annabel for lunch, but Annabel lies in Pittsburg, 400 miles East of Chicago. If George leaes home at 9 am traeling 60 mph East and Susan leaes at the same time traeling 40 mph West, where will they meet? 1) ical Representation - Motion Diagram George Chicago a George Annabel a Annabel Pittsburg George lies in Chicago and want to meet up with his friend Annabel for lunch, but Annabel lies in Pittsburg, 400 miles East of Chicago. If George leaes home at 9 am traeling 60 mph East and Susan leaes at the same time traeling 40 mph West, where will they meet? 2) Pictorial Representation Unknown:, x f G x x G,i t i a G Known: t i, x A,f = x G,f = x f, x G,i, x A,i = +400 miles G = +60 mph, A = - 40 mph, a G, a A A x A,f x G,f G a A A x A,i t i +x George lies in Chicago and want to meet up with his friend Annabel for lunch, but Annabel lies in Pittsburg, 400 miles East of Chicago. If George leaes home at 9 am traeling 60 mph East and Susan leaes at the same time traeling 40 mph West, where will they meet? Kinematic equation for George: x G, f = x G,i + G,i ( t) + 1 a 2 G( t) 2 + (60mph)( ) + 0 = (60mph) Kinematic equation for Annabel: x A, f = x A,i + A,i ( t) a A ( t)2 George lies in Chicago and want to meet up with his friend Annabel for lunch, but Annabel lies in Pittsburg, 400 miles East of Chicago. If George leaes home at 9 am traeling 60 mph East and Susan leaes at the same time traeling 40 mph West, where will they meet? The condition for meeting at time is that x G,f = x A,f, so we need to set the two kinematic equations from the last slide equal to one another. x A, f = 400miles (40mph) = (60mph) = x G, f (100mph) = 400miles = 4hours = 400miles + ( 40mph)( ) + 0 = 400miles (40mph)

8 George lies in Chicago and want to meet up with his friend Annabel for lunch, but Annabel lies in Pittsburg, 400 miles East of Chicago. If George leaes home at 9 am traeling 60 mph East and Susan leaes at the same time traeling 40 mph West, where will they meet? So, they ll meet up at 1 pm, but the question wants to know where. To find out, we can use either of the kinematic equations, plugging in our answer for. x A, f = 400miles (40mph)(4hours) = 240miles George lies in Chicago and want to meet up with his friend Annabel for lunch, but Annabel lies in Pittsburg, 400 miles East of Chicago. If George leaes home at 9 am traeling 60 mph East and Susan leaes at the same time traeling 40 mph West, where will they meet? 4) Sanity Check It s useful to ask yourself if the answer makes sense. We know that George was traeling with a higher speed. We also know that they both left at the same time. That means that when they meet up, they ll both hae been traeling for the same amount of time. So, George must hae gone further than Annabel. Indeed, our answer of 240 miles East of Chicago is farther from Chicago than Pittsburg. Good deal! Please note on your worksheet the meaning of the LENGTH of the line in the following ersus time graphs: Class Worksheet #6: Reiew x s t