RENE DESCARTES ( ) Motion In Two Dimensions GALILEO GALILEI ( )

Vectors in Physics A scalar quantity has only magnitude. All physical quantities are either scalars or vectors A vector quantity has both magnitude and direction. Scalars Vectors Common examples are length, area, volume, time, mass, energy, voltage, and temperature. Common examples are acceleration, force, electric field, momentum. In kinematics, distance and speed are scalars. In kinematics, position, displacement, and velocity are vectors.

Representing Vectors The arrow’s length represents the vector’s magnitude A simple way to represent a vector is by using an arrow. The arrow’s orientation represents the vector’s direction “Standard Angle” “Bearing Angle”  0˚  90˚ 180˚ 270˚ E, 90˚ N, 0˚ W, 270˚ S, 180˚ In physics, a vector’s angle (direction ) is called “theta” and the symbol is θ. Two angle conventions are used:

Vector Math Vector Equivalence Two vectors are equal if they have the same length and the same direction. Two vectors are opposite if they have the same length and the opposite direction. Vector Opposites equivalence allows vectors to be translated opposites allows vectors to be subtracted

click for applet Head to Tail Addition Vectors add according to the “Head to Tail” rule. The tail of a vector is placed at the head of the previous vector. The resultant vector is from the tail of the first vector to the head of the last vector. (Note that the resultant itself is not head to tail.) For the Vector Field Trip, the resultant vector is 68.6 meters, 79.0˚ South Lawn Vector Walk click for web site

Graphical Addition of Vectors Vector Addition Vectors add according to the “Head to Tail” rule. The resultant vector isn’t always found with simple arithmetic! click for applet click for applet simple vector addition right triangle vector addition non-right triangle vector addition Vector Subtraction To subtract a vector simply add the opposite vector. simple vector subtraction right triangle vector subtraction

Resolving Vectors It is useful to resolve or “break down” vectors into component vectors. (Same as finding rectangular coordinates from polar coordinates in math.)

Finding a Resultant Often a vector’s components are known, and the resultant of these components must be found. (Same as finding polar coordinates from rectangular coordinates in math.) Finding the magnitude of the resultant: Finding the direction of the resultant:

Resolving Vectors Example: A baseball is thrown at 30 m/s at an angle of 35˚ from the ground. Find the horizontal and vertical components of the baseball’s velocity. Horizontal velocity: Vertical velocity:

Finding a Resultant Example: A model rocket is moving forward at 10 m/s and downward at 4 m/s after it has reached its peak. What overall velocity (magnitude and direction) does the rocket have at this moment? Find the rocket’s speed (magnitude) Find the rocket’s angle (direction)

Projectile Motion vxvx vxvx vxvx vxvx vxvx vyvy vyvy vyvy vyvy vxvx vyvy vyvy vxvx vxvx vyvy v v Horizontal: constant motion, a x = 0 Vertical: freefall motion, a y = g = 9.8 m/s 2 velocity is tangent to the path of motion click for applet click for applet Projectile motion = constant motion + freefall motion θ resultant velocity: v vyvy

Projectile Motion at an Angle click for applet click for applet velocity components: vxvx vxvx vxvx vxvx vxvx vxvx vxvx vxvx vxvx θ vyvy vyvy vyvy vyvy vyvy vyvy vyvy vyvy vxvx θ v yi v vertical velocity, v y is zero here! v v v v v v v v

Relative Velocity All velocity is measured from a reference frame (or point of view). Velocity with respect to a reference frame is called relative velocity. A relative velocity has two subscripts, one for the object, the other for the reference frame. Relative velocity problems relate the motion of an object in two different reference frames. refers to the object refers to the reference frame click for reference frame applet click for relative velocity applet velocity of object a relative to reference frame b velocity of reference frame b relative to reference frame c velocity of object a relative to reference frame c

Relative Velocity At the airport, if you walk on a moving sidewalk, your velocity is increased by the motion of you and the moving sidewalk. v pg = velocity of person relative to ground v ps = velocity of person relative to sidewalk v sg = velocity of sidewalk relative to ground When flying against a headwind, the plane’s “ground speed” accounts for the velocity of the plane and the velocity of the air. v pe = velocity of plane relative to earth v pa = velocity of plane relative to air v ae = velocity of air relative to earth

Relative Velocity When flying with a crosswind, the plane’s “ground speed” is the resultant of the velocity of the plane and the velocity of the air. v pe = velocity of plane relative to earth v pa = velocity of plane relative to air v ae = velocity of air relative to earth Sometimes the vector sums are more complicated! Pilots must fly with crosswind but not be sent off course.