# Conceptual Physics Motion and Graphs Free Fall Using Vectors

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1 Conceptual Physics Motion and Graphs Free Fall Using Vectors Lana heridan De Anza College July 6, 2017

2 Last time Units More about size and scale Motion of objects Inertia Quantities of motion

3 Overview graphs of motion with time free fall Vectors relative motion

4 peed, velocity, acceleration peed is the rate of distance covered with time. Velocity, v, is speed with direction specified. Acceleration, a, is the rate of change of velocity with time.

5 Graphs and Physics Graphs represent the values of a function (eg. f (x)) as a variable changes (eg. x). In physics, the function and the variable are physical quantities: things we can measure, eg. position and time. The slope tells us about the rate that one quantity changes when we change the other.

6 Table 2.1 Position of the Car at Various Times Position t (s) x (m) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 50 m to the left of x 5 0 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. uch a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- The car moves to the right between positions and. x (m) x (m) a The car moves to the left between positions and. x (m) t b x t (s) Figures from erway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. everal representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

7 Average Velocity in Position vs. Time Graphs Chapter 2 Motion in One Dimension Green line: slope is the instantaneous velocity at point A. x (m) t (s) 50 Blue lines: slopes a are the average velocities from A B and A E. b Figure 2.3 (a) Graph representing the motion of the car in F

8 Average Velocity in Position vs. Time Graphs Chapter 2 Motion in One Dimension A B: v avg = x t = 50m 30m 10s 0s = 2 m/s x (m) A E: v a avg = x t = 35m 30m 40s 0s = 1.6 m/s 40 t (s) Figure 2.3 (a) Graph representing the motion of the car in F b

9 Relating Position, Velocity, Acceleration graphs For a single moving object, the graphs of its position, velocity, and acceleration are not independent! The slope of the position-time graph is the velocity. The slope of the velocity-time graph is the acceleration.

10 Constant Acceleration Graphs x i x lope v xf x lope v xi lope v xf t x i alope v xi x i vlope x v a xi lope a x v a x v xi lope a x v x v xi lope a x b v xi ab x a x b a x t va xi x t v xi v xf t t lope 0 lope 0 lope 0 t t t t t a x t t v xi a x t a x a x v xf v xf t t t t 2.6 Analysis Mod If the acceleration of a parti Under to analyze. 2.6 Analysis Constant A very common Mod Ac that in which the accelerat aunder x,avg over any Constant time interval A If the acceleration of a particl to analyze. A very common a at any instant within the int that If the in acceleration which the acceleratio out the motion. This of a situat partic ato x,avg analyze. over any A time very common interval isa model: that any in instant the particle which within the acceleratio the under inter out generate a x,avg the over motion. several any time This equations interval situatioi model: If we at any instant the replace particle a within x,avg under by a the inte co x i generate t, we find out the motion. several that equations This situatio th model: If we the replace particle a x,avg under by a x inc t, generate we find several that equations th or If we replace a x,avg by a x in t, we find that v or This powerful expression v e xf or t if we know the object s This velocity time powerful expression graph for thi en v t The if we graph know is the a straight object s line xf velocity time This slope powerful is consistent graph expression with for this a x en 5c The t tive, if we graph which know is indicates a the straight object s a line, pos in t velocity time slope is of consistent the line graph in with for Figure a this x 5 tive, The stant, which graph the indicates graph is a straight of a acceler positi line,

11 Acceleration vs. Time Graphs angent nd t 5 mly, so e 2.8b. is cont and alue of e slope at that graph se unie slope o zero. mean- a b x v x a x t t t t t t t t t t t t t t he tanf accelre 2.8c. 0 and c t t t t t

12 Acceleration and Free Fall Galileo also reasoned about the acceleration due to gravity by thinking more about inclined surfaces. h θ O v The steeper the incline the larger the acceleration. tarting from rest: final velocity = acceleration time.

13 Free Fall When the ball drops straight downward, it gains approximately 10 m/s of speed in each second. Time of fall (s) Velocity acquired (m/s) This is a constant acceleration! We call this acceleration g.. g = 9.8 m s 2 10 m s 2 g is about 10 meters-per-second-per-second

14 Free Fall Constant acceleration corresponds to a straight line on a graph of velocity and time. Calling up positive and down negative: v = v i gt Object s velocity changes by 10 m/s each second. When dropped from rest, after 6.5 s, (roughly) what is the ball s speed?

15 Free Fall Constant acceleration corresponds to a straight line on a graph of velocity and time. Calling up positive and down negative: v = v i gt Object s velocity changes by 10 m/s each second. When dropped from rest, after 6.5 s, (roughly) what is the ball s speed? 65 m/s

16 Acceleration due to gravity 65 m/s is very fast. ( 145 mi/hr) We rarely see falling objects going this fast. (Why?)

17 Acceleration due to gravity 65 m/s is very fast. ( 145 mi/hr) We rarely see falling objects going this fast. (Why?) How far does the ball travel in those 6.5 s (ignoring air resistance)? distance an object falls (starting from rest) in time t: d = 1 2 gt2 This corresponds to the area under the velocity-time graph.

18 Falling Objects CHAPTER 2 KINEMATIC 65 Figure 2.40 Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly with time and that acceleration is constant. Misconception Alert! Notice that the position vs. time graph shows vertical position only. It is easy to get the impression that the graph shows some horizontal motion the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is time, not space. The 1 Opentax Physics

19 Free Fall Questions A free-falling object has a speed of 30 m/s (downward) at one instant. Exactly 1 s later its speed will be A the same. B 35 m/s. C more than 35 m/s. D 60 m/s.

20 Free Fall Questions A free-falling object has a speed of 30 m/s (downward) at one instant. Exactly 1 s later its speed will be A the same. B 35 m/s. C more than 35 m/s. D 60 m/s.

21 Free Fall Questions A free-falling object has a speed of 30 m/s (downward) at one instant. Exactly 1 s later its speed will be A the same. B 35 m/s. C more than 35 m/s. D 60 m/s. It will be 40 m/s. (Assuming g = 10 m/s 2 )

22 Free Fall Questions What is the distance covered by a freely falling object starting from rest after 4 s? A 4 m B 16 m C 40 m D 80 m

23 Free Fall Questions What is the distance covered by a freely falling object starting from rest after 4 s? A 4 m B 16 m C 40 m D 80 m

24 Review questions for motion Can an object reverse its direction of travel while maintaining a constant acceleration? 1 Hewitt, page 49.

25 Review questions for motion Can an object reverse its direction of travel while maintaining a constant acceleration? You are driving north on a highway. Then, without changing speed, you round a curve and drive east. Did you accelerate? 1 Hewitt, page 49.

26 Review questions for motion Can an object reverse its direction of travel while maintaining a constant acceleration? You are driving north on a highway. Then, without changing speed, you round a curve and drive east. Did you accelerate? Neglecting the effect of air resistance, how does the acceleration of a ball that has been thrown straight up compare with its acceleration if simply dropped? 1 Hewitt, page 49.

27 Review questions for motion Can an object reverse its direction of travel while maintaining a constant acceleration? You are driving north on a highway. Then, without changing speed, you round a curve and drive east. Did you accelerate? Neglecting the effect of air resistance, how does the acceleration of a ball that has been thrown straight up compare with its acceleration if simply dropped? Why does a stream of water get narrower as if falls from a faucet? 1 Hewitt, page 49.

28 Review questions for motion Can an object reverse its direction of travel while maintaining a constant acceleration? You are driving north on a highway. Then, without changing speed, you round a curve and drive east. Did you accelerate? Neglecting the effect of air resistance, how does the acceleration of a ball that has been thrown straight up compare with its acceleration if simply dropped? Why does a stream of water get narrower as if falls from a faucet? Is the acceleration due to gravity always 9.8 m s 2? 1 Hewitt, page 49.

29 Describing Vectors: Axes To indicate which way an arrow (or a force, acceleration, etc.) points, we need to have another arrow that we can compare to.

30 Describing Vectors: Axes To indicate which way an arrow (or a force, acceleration, etc.) points, we need to have another arrow that we can compare to. For example, North-outh and West-East can be reference axes.

31 Describing Vectors: Axes To indicate which way an arrow (or a force, acceleration, etc.) points, we need to have another arrow that we can compare to. For example, North-outh and West-East can be reference axes. We could also choose axes up and down (vertical), and parallel to the horizon (horizontal). We typically call the direction axes x and y.

32 ght triangle and that A 5 A x 1 A y. We shall often refer to the component a Vectors vector A, written A x and A y (without the boldface notation). The compo nt A x represents the projection of A along the x axis, and the component A presents the projection of A along the y axis. These components can be positiv negative. To describe The component where the A vector x is positive A points, if the we component can say, you vector count some A x points i e positive x direction and is negative if A x points in the negative x direction. distance along the x direction, the some distance along the y ilar statement is made for the component A direction. y. y y A A A y Ay O a u Ax x O b u Ax x 1 Figure from erway & Jewett, Physics for cientists and Engineers, 9th ed.

33 Representing Vectors: Unit Vectors We can count along the directions with special units: unit vectors. Unit vectors have a length of one unit.

34 Representing Vectors: Unit Vectors We can count along the directions with special units: unit vectors. Unit vectors have a length of one unit. A unit vector in the x direction is usually written i A unit vector in the y direction is usually written j.

35 parallel to the y axis. From Figure 3.12b, we see that the three vectors form ght triangle and that Components A 5 A x 1 A y. We shall often refer to the component a vector A, written A x and A y (without the boldface notation). The compo nt A x represents the projection of A along the x axis, and the component A presents Vector the projection A is the sum of of A along piecethe along y axis. x and These a piece components along y: can be positiv negative. The component A x is positive if the component vector A x points i e positive x direction and is negative A = Aif x i + A A y j. x points in the negative x direction. ilar statement is made for the component A y. x component y component y y A A A y Ay O a u Ax x O b u Ax x Notice that A x = A cos θ and A y = A sin θ.

36 Adding Vectors Commutative law of addition construction in Figure 3.8, is known as the commut A 1 B 5 B 1 A A + B A R A B B D B C A A D B C Figure 3.6 When vector B is added to vector A, the resultant R is the vector that runs from the tail of A to the tip of B. Figure 3.7 Geometric construction for summing four vectors. The resultant vector R is by definition the one that completes the polygon.

38 Using Vectors Example Andy runs 100 m south then turns west and runs another 50.0 m. All this takes him 15.0 s. What is his displacement from his starting point? What is his average velocity?

39 Using Vectors Example Andy runs 100 m south then turns west and runs another 50.0 m. All this takes him 15.0 s. What is his displacement from his starting point? What is his average velocity? answer: displacement: 112 m average velocity: 7.45 m/s, in a direction 26.6 west of south.

40 Thinking about Vectors What can you say about two vectors that add together to equal zero?

41 Thinking about Vectors What can you say about two vectors that add together to equal zero? When can a nonzero vector have a zero horizontal component?

42 Using Vectors We ve seen how to add vectors, but what do we use this for? We ll look at examples. Finding components of velocity. Relative motion. Finding net force.

43 nts of Vectors Components of Velocity can be resolved into horizontal and ents. If we know the vertical component of velocity, we can find the time of flight, the maximum height, etc. of the ball. 1 Drawing by Hewitt, via Pearson.

44 Acceleration due to gravity and kinematics nts of Vectors Let s think about the components of the motion separately. can be resolved into horizontal and ents. Vertical (y-direction): v y = v i,y gt d y = v i,y t 1 2 gt2 Horizontal (x-direction): a x = 0, v x = v i,x d x = v i,x t We will return to this in a later chapter! 1 Drawing by Hewitt, via Pearson.

45 Linear Motion When we say something is moving, we mean that it is moving relative to something else. Motion is relative. In order to describe measurements of where something is how fast it is moving we must have reference frames. An example of references for time an space might be picking an object, declaring that it is at rest, and describing the motion of all objects relative to that.

46 Intuitive Example for Relative Velocities tors ocity relative to ds on the relative to the d s velocity.

47 Intuitive Example Now, imagine an airplane that is flying North at 80 km/h but is blown off course by a cross wind going East at 60 km/h. How fast is the airplane moving relative to the ground? In which direction?

48 Intuitive Example Now, imagine an airplane that is flying North at 80 km/h but is blown off course by a cross wind going East at 60 km/h. 5 Projectile Motion How fast is 5.2 thevelocity airplane Vectors moving relative to the ground? In which direction? An 80-km/h airplane flying in a 60-km/h crosswind has a resultant speed of 100 km/h relative to the ground.

49 Intuitive Example Now, imagine an airplane that is flying North at 80 km/h but is blown off course by a cross wind going East at 60 km/h. 5 Projectile Motion How fast is 5.2 thevelocity airplane Vectors moving relative to the ground? In which direction? An 80-km/h airplane flying in a 60-km/h crosswind has a resultant speed of 100 km/h relative to the ground. v = 100 km/h at 36.9 East of North (or 53.1 North of East)

50 Relative Motion and Components of Velocity Hewitt, page 79, ranking question 4. Three motorboats crossing a river.

51 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h frame relative to the water. The water B, which moves to the right with a constant in the velocity river v BA. has The a uniform speed of 5.00 km/h due east relative vector rthe PA is the Earth. particle s If position the boat heads due vector relative to A, and r P B is its north, determine the velocity position of the vector boat relative relative to B. to an observer standing on either bank. 1 ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe W b N E vbr v re 2 Page Figure 97, 4.21 erway (Example & Jewett 4.8) (a) A boat aims directly across a u vbe

52 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h frame relative to the water. The water B, which moves to the right with a constant in the velocity river v BA. has The a uniform speed of 5.00 km/h due east relative vector rthe PA is the Earth. particle s If position the boat heads due vector relative to A, and r P B is its north, determine the velocity position of the vector boat relative relative to B. to an observer standing on either bank. 1 v br = 10.0 km/h v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe W b N E vbr v re 2 Page Figure 97, 4.21 erway (Example & Jewett 4.8) (a) A boat aims directly across a u vbe

53 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h frame relative to the water. The water B, which moves to the right with a constant in the velocity river v BA. has The a uniform speed of 5.00 km/h due east relative vector rthe PA is the Earth. particle s If position the boat heads due vector relative to A, and r P B is its north, determine the velocity position of the vector boat relative relative to B. to an observer standing on either bank. 1 v br = 10.0 km/h v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe W b N E vbr v re 2 Page Figure 97, 4.21 erway (Example & Jewett 4.8) (a) A boat aims directly across a imply use vector addition to find v be. u vbe v be = = 11.2 km/h

54 Relative Motion Example v BA dt Figure 4.20 A particle located at P is described by two observers, one in the fixed frame of refer- A boat crossing a wide river ence moves A and the with other ain speed the of 10.0 km/h frame relative to the water. The water B, which moves to the right with a constant in the velocity river v BA. has The a uniform speed of 5.00 km/h due east relative vector rthe PA is the Earth. particle s If position the boat heads due vector relative to A, and r P B is its north, determine the velocity position of the vector boat relative relative to B. to an observer standing on either bank. 1 v br = 10.0 km/h v re = 5.00 km/h ore, we conclude that a P A 5 a P B hat is, the acceleration of the partirence is the same as that measured city relative to the first frame. r peed of ver has a e Earth. locity of bank. s a river will not end up W a N vbr E you relau vre vbe W b N E vbr v re 2 Page Figure 97, 4.21 erway (Example & Jewett 4.8) (a) A boat aims directly across a imply use vector addition to find v be. u vbe v be = = 11.2 km/h ( ) 5 θ = tan 1 =

55 ummary graphing motional quantities free fall vectors Homework Worksheets, graphs worksheet (for Tues) vector worksheet (for Tues) Hewitt, Ch 3, onward from page 47 Review Questions: 3, 9, 21 Plug and Chug: 5, 13, 21 Exercises: 15, 17, 19, 41 read pages 74-77, in Ch 5 Ch 5, onward from page 78. Exercises: 31, 33

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