Chapter 3 Motion in a Plane MFMc GrawPHY

Chapter 3 Motion in a Plane MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 1

Motion in a Plane • Vector Addition • Velocity • Acceleration • Projectile motion MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 2

Graphical Addition and Subtraction of Vectors A vector is a quantity that has both a magnitude and a direction. Position is an example of a vector quantity. A scalar is a quantity with no direction. The mass of an object is an example of a scalar quantity. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 3

Notation Vector: The magnitude of a vector: The direction of vector might be “ 35 south of east”; “ 20 above the +x-axis”; or…. Scalar: m (not bold face; no arrow) MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 4

Graphical Addition of Vectors To add vectors graphically they must be placed “tip to tail”. The result (F 1 + F 2) points from the tail of the first vector to the tip of the second vector. This is sometimes called the resultant vector R F 2 R F 1 MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 5

Vector Simulation MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 6

Examples • Trig Table • Vector Components • Unit Vectors MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 7

Types of Vectors MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 8

Relative Displacement Vectors Vector Addition Vector Subtraction is a relative displacement vector of point P 3 relative to P 2 MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 9

Vector Addition via Parallelogram MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 10

Graphical Method of Vector Addition MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 11

Graphical Subtraction of Vectors Think of vector subtraction A B as A+( B), where the vector B has the same magnitude as B but points in the opposite direction. Vectors may be moved any way you please (to place them tip to tail) provided that you do not change their length nor rotate them. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 12

Vector Components MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 13

Vector Components MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 14

Graphical Method of Vector Addition MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 15

Unit Vectors in Rectangular Coordinates MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 16

Vector Components in Rectangular Coordinates MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 17

Vectors with Rectangular Unit Vectors MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 18

Dot Product - Scalar The dot product multiplies the portion of A that is parallel to B with B MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 19

Dot Product - Scalar In 2 dimensions In any number of dimensions The dot product multiplies the portion of A that is parallel to B with B MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 20

Cross Product - Vector The cross product multpilies the portion of A that is perpendicular to B with B MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 21

Cross Product - Vector In 2 dimensions In any number of dimensions MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 22

Velocity MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 23

A particle moves along the curved path as shown. At time t 1 its position is ri and at time t 2 its position is rf. y vi r ri vf The instantaneous velocity points tangent to the path. rf x Points in the direction of r MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 24

A displacement over an interval of time is a velocity The instantaneous velocity is represented by the slope of a line tangent to the curve on the graph of an object’s position versus time. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 25

Acceleration MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 26

A particle moves along the curved path as shown. At time t 1 its position is r 0 and at time t 2 its position is rf. y vi Points in the direction of v. v vf ri rf x MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 27

A nonzero acceleration changes an object’s state of motion These have interpretations similar to vav and v. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 28

Motion in a Plane with Constant Acceleration - Projectile What is the motion of a struck baseball? Once it leaves the bat (if air resistance is negligible) only the force of gravity acts on the baseball. Acceleration due to gravity has a constant value near the surface of the earth. We call it g = 9. 8 m/s 2 Only the vertical motion is affected by gravity MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 29

Projectile Motion The baseball has ax = 0 and ay = g, it moves with constant velocity along the x-axis and with a changing velocity along the yaxis. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 30

Example: An object is projected from the origin. The initial velocity components are vix = 7. 07 m/s, and viy = 7. 07 m/s. Determine the x and y position of the object at 0. 2 second intervals for 1. 4 seconds. Also plot the results. Since the object starts from the origin, y and x will represent the location of the object at time t. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 31

Example continued: MFMc. Graw-PHY 1401 t (sec) x (meters) y (meters) 0 0. 2 1. 41 1. 22 0. 4 2. 83 2. 04 0. 6 4. 24 2. 48 0. 8 5. 66 2. 52 1. 0 7. 07 2. 17 1. 2 8. 48 1. 43 1. 4 9. 89 0. 29 Chapter 3 b - Revised: 6/7/2010 32

Example continued: This is a plot of the x position (black points) and y position (red points) of the object as a function of time. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 33

Example continued: This is a plot of the y position versus x position for the object (its trajectory). The object’s path is a parabola. MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 34

Example (text problem 3. 50): An arrow is shot into the air with = 60° and vi = 20. 0 m/s. (a) What are vx and vy of the arrow when t = 3 sec? y The components of the initial velocity are: vi 60° x CONSTANT At t = 3 sec: MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 35

Example continued: (b) What are the x and y components of the displacement of the arrow during the 3. 0 sec interval? y r x MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 36

Example: How far does the arrow in the previous example land from where it is released? The arrow lands when y = 0. Solving for t: The distance traveled is: MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 37

Summary • Adding and subtracting vectors (graphical method & component method) • Velocity • Acceleration • Projectile motion (here ax = 0 and ay = g) MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 38

Projectiles Examples • Problem solving strategy • Symmetry of the motion • Dropped from a plane • The home run MFMc. Graw-PHY 1401 Chapter 3 b - Revised: 6/7/2010 39