PES 1110 Fall 2013, Spendier Lecture 5/Page 1

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1 PES 1110 Fall 2013, Spendier Lecture 5/Page 1 Toda: - Announcements: Quiz moved to net Monda, Sept 9th due to website glitch! - Finish chapter 3: Vectors - Chapter 4: Motion in 2D and 3D (sections ) - Questions on HW 1? - net week Monda: Projectile Motion and Quiz 1! Review last time: Vectors - Vectors have a magnitude and a direction - Draw vectors: use standard angle from the positive -ais (counterclockwise, CCW) - Vector addition: tip to tail (pictoriall) and b components A ˆ = Ai A ˆ = Aj A= A + A = Aiˆ + A ˆj A = Acosθ A = Asinθ magnitude: A = A + direction: θ 2 ( ) ( A ) 2 1 = tan A A We use the smbols î, ĵ, and ˆk for the unit vectors along the,, and z aes. Adding vectors using components: A+ B= A + B i ˆ + A + B ˆ j+ A + B k ( ) ( ) ( ) ˆ z z Eamples: Using unit vectors Given the displacements D= ( 6i ˆ + 3 ˆ j k ˆ ) m magnitude of the displacement 2D E E i j k m. Find the and = ( 4 ˆ 5 ˆ + 8 ˆ ) We are first multipling the vector D b 2 (a scalar) and then subtract the vector E from the result. F= 2D E= 2 6i+ 3j k m 4i 5j+ 8k m ( ˆ ˆ ˆ) ( ˆ ˆ ˆ) 1

2 PES 1110 Fall 2013, Spendier Lecture 5/Page 2 F= ( 12iˆ+ 6 ˆj 2kˆ) m ( 4iˆ 5 ˆj+ 8kˆ) m F= (12 4) iˆ+ (6+ 5) ˆj+ ( 2 8) kˆ m F= 8iˆ+ 11ˆj 10kˆ m magnitude: F= F + F + F = ( 10) =17 m z Two-Dimensional Motion Moving objects in two dimensions, everthing is vectors; everthing is components position, velocit, and acceleration In 2D - It boils down to taking track of twice as man equations and twice as man variables to keep track of, because we must keep track of all the components we just talked about. To describe motion, we still need to know position, velocit, and acceleration at all times. In 2D, this means we have to know the components of the position, velocit, and acceleration vectors. To locate an object, we have to give two numbers: (, ). The are the Cartesian coordinates AND the are the components of the position vector. Position Vector: In Phsics components have real and useful meaning. In phsics it is all about components: how do ou locate something in two dimensions? You give two numbers of and. Eample: What are the components of the position vector in 2d? The components are the a and Cartesian coordinates: = how far to the right or left of the origin = how far up or down of the origin is the position? We have the same procedure for finding components: 2

3 PES 1110 Fall 2013, Spendier Lecture 5/Page 3 = position vector s -component (horizontal motion) = position vector s -component (vertical motion) r = position vector from the origin (tip to tail addition) In unit-vector notation: 2D: r = + = î + ĵ 3D: r = + = î + ĵ + z ˆk Eample: Draw a particle with position vector r = (-3m)î + (5m)ĵ Along the -ais the particle is 3 m from the origin in the minus î direction Along the -ais the particle is 5 m from the origin in the plus ĵ direction Displacement Vector A real application of all this we just learned adding and subtracting vectors is the displacement vector. The displacement vector r= r 2 r is a vector subtraction: 1 the final position vector r 2 minus the initial position vector r 1 r points from to r 1 to r 2 ( what the motion does in between we do not know in this case) Position vectors are from the origin. 3

4 PES 1110 Fall 2013, Spendier Lecture 5/Page 4 Eample: A hiker walks 10.0 km west and then 7.50 km south. What is the hikers displacement (magnitude and direction)? W N (+) E (+) = θ S Magnitude: = 10 = 7.5 ( ) ( ) ( ) ( ) magnitude : A = + = = = 12.5km Direction: o 7.5 tanθ = = a θ = tan = use relative bearing: 90º - θ = 90º-36.9º = 53.1 º θ α Answer: The hiker s displacement is 12.5 km at 53.1 º west of south. Eample: An old man walks on three different bearings: 72.4 m, 32.0º east of north 57.3 m, 36.0 º south of west 17.8 m straight south What is the man's displacement at the end of his walk? Find the sum (resultant) of the three displacements. a) draw a picture 4

5 PES 1110 Fall 2013, Spendier Lecture 5/Page 5 b) find vector components: The angles of the vectors, measured from the +-ais toward the +-ais are 90.0 º 32.0 º = 58.0 º 180 º º = 216 º 270 º then A = Acosθ A = (74.4 m)(cos 58.0 º) = m A = Asinθ A = (74.4 m)(sin 58.0 º) = m B = Bcosθ B = (57.3 m)(cos 216 º) = m B = Bsinθ B = (57.3 m)(sin 216 º) = m C = Ccosθ C = (17.8 m)(cos 270 º) = 0.00 m C = Csinθ C = (17.8 m)(sin 270 º) = m R = A + B + C = m + ( m) m = m R = A + A + C = m + ( m) m = 9.92 m R= R + R = ( 7.99) = 12.7m R θ= tan tan 129 R = = 7.99 Or 129º 90º = 39º west of north Average velocit vector Let s think about the average velocit vector. It is in the same direction as displacement vector. If it is actuall longer or shorter we don t know at the moment. The average velocit is parallel to r : r ˆ ˆ ˆ ˆ ˆ ˆ 2 1 i+ j+ zk i j zk vavg= = = = + + t t t t t t 5

6 PES 1110 Fall 2013, Spendier Lecture 5/Page 6 Eample: What is the resultant velocit of a plane that is heading north at 290 m/s when also aided b a wind of 46.0 m/s NW? 6

7 PES 1110 Fall 2013, Spendier Lecture 5/Page 7 Trajector: Picture of the actual path taken b an object. A trajector tells ou where the object was located at all the different points of motion. There are two separate position plots which give the velocit vector s components At ever point we measure its value of and Both of these graphs of course have slopes. Slope of blue graph gives me the component of the velocit, v. The green graph gives me the component of the velocit vector, v. Two different positions two different slopes neither gives me the full stor of the component. When we do make this trajector graph, the velocit components v and v the will make a vector which is tangent to the trajector - this is the velocit vector. The velocit vector is at the same angle as the slope of the trajector graph. Instantaneous Velocit vector dr d ( ˆ ˆ ˆ d v= ) ˆ d ˆ dz = i + j+ zk = i + j+ kˆ dt dt dt dt dt v=v ˆi+v ˆj+v kˆ z d d dz v = ; v = ; v z = ; dt dt dt For eample with differentiation look at book: page 62 The velocit vector will go in one direction how do we determine which wa it should go? The orange line has as arrow on it. We use arrows to indicate the direction the arrow on the trajector line will indicate which direction the velocit vector should point. 7

8 PES 1110 Fall 2013, Spendier Lecture 5/Page 8 Technicall speaking I should not mi velocities with positions. Because the are completel different quantities, completel different units. We reall should not be drawing them on the same graph. The directions and length of the components are good. Just don t compare the length of v to Speed: Speed is just how fast the speed at this point how fast is it going, how big is the velocit vector, speed is the magnitude of the velocit vector. From the and components of the velocit vector use the Pthagorean Theorem (speed is v with no arrow) speed = v = v + v 2 2 Average acceleration vectors: The average velocit is parallel to v v v ˆ ˆ ˆ ˆ ˆ ˆ 2 v vi 1 + vj+ vk z vi v j vk z aavg= = = = + + t t t t t t : Instantaneous acceleration vector: In 2D I can make two separate velocit plots, give the components of the acceleration, two plots v and v both functions of time, and components versus time. We can find the acceleration components in the same wa as velocit There are two separate VELOCITY plots which give the acceleration vector s components Slopes will give me the or component of the acceleration vector: Instantaneous acceleration 8

9 PES 1110 Fall 2013, Spendier Lecture 5/Page 9 Find tangent line at each - point on trajector, find slope of tangent get v and v at a given time point - use this to make a velocit vs time graph where the slope will give me the and components of the acceleration. v. how fast is it traveling verticall v. how fast is it traveling horizontall vertical component of position gives me vertical component of velocit which gives me vertical component of acceleration! dv d ( ˆ ˆ ˆ dv dv a= ) ˆ ˆ dvz = vi ˆ + v j + vk z = i + j + k dṱ dt dt dt dt a=a i+a ˆj+a kˆ z dv dv dvz v = ; v = ; v z = ; dt dt dt For eample with differentiation look at book: page 63 9

10 PES 1110 Fall 2013, Spendier Lecture 5/Page 10 Eample: What is the average acceleration of a plane that changes its velocit from 350m/s south to 350m/s NW in a time of 14.00s? 10