Motion In Two Dimensions. Vectors in Physics

Transcription

1 Motion In Two Dimensions RENE DESCARTES ( ) GALILEO GALILEI ( ) Vectors in Physics All physical quantities are either scalars or ectors Scalars A scalar quantity has only magnitude. In kinematics, time, distance and speed are scalars. Other examples: length, mass, power. Some are een negatie (charge, energy, oltage, and temperature) but not directional. Vectors A ector quantity has both magnitude and direction. In kinematics, position, displacement, and elocity, and acceleration are ectors. Other examples: forces, fields (electric, magnetic, graitational), and momentum. 1

2 Representing Vectors An arrow is a simple way to represent a ector. The arrow s length represents the ector s magnitude The arrow s orientation represents the ector s direction In physics, a ector s angle (direction ) is called theta and the symbol is often. Two angle conentions are used: 90 N, Standard Angle 0 W, 270 Bearing Angle E, S, 180 Vector Math Vector Equialence Two ectors are equal if they hae the same length and the same direction. equialence allows ectors to be translated Vector Opposites Two ectors are opposite if they hae the same length and the opposite direction. opposites allows ectors to be subtracted 2

3 Vector Addition Graphical Addition of Vectors Vectors add according to the Head to Tail rule. The resultant ector isn t always found with simple arithmetic! simple ector addition right triangle ector addition Vector Subtraction To subtract a ector simply add the opposite ector. non-right triangle ector addition simple ector subtraction non-right triangle ector subtraction web site Head to Tail Addition Vectors add according to the Head to Tail rule. The tail of a ector is placed at the head of the preious ector. The resultant ector is from the tail of the first ector to the head of the last ector. (Note that the resultant itself is not head to tail.) For the Vector Field Trip, the resultant ector is 69.9 meters, 78.0 South Lawn Vector Walk web site 3

4 Resoling Vectors, Finding Resultant To resole a ector into component ectors, use trigonometry: Finding the horizontal component Finding the ertical component If the ector components are known, the resultant can be found: Finding the resultant s direction Finding the resultant s magnitude The ector components are rectangular coordinates (x,y) The ector magnitude & direction are polar coordinates (r,) Example of Vector Addition Honors: 4

5 Projectile Motion Horizontal Launch Vertical: freefall motion, a y = g = 9.8 m/s 2 Horizontal: constant motion, a x = 0 Projectile motion = constant motion + freefall motion elocity is tangent to the path of motion resultant elocity: Projectile Motion Non Zero Launch Angle ertical elocity, is zero here! i elocity components: 5

6 Relatie Velocity All elocity is measured from a reference frame (or point of iew). Velocity with respect to a reference frame is called relatie elocity. A relatie elocity has two subscripts, one for the object, the other for the reference frame. Relatie elocity problems relate the motion of an object in two different reference frames. reference frame refers to the object relatie elocity refers to the reference frame elocity of object a relatie to reference frame b elocity of reference frame b relatie to reference frame c elocity of object a relatie to reference frame c Relatie Velocity At the airport, if you walk on a moing sidewalk, your elocity is increased by the motion of you and the moing sidewalk. pg = elocity of person relatie to ground ps = elocity of person relatie to sidewalk sg = elocity of sidewalk relatie to ground When flying against a headwind, the plane s ground speed accounts for the elocity of the plane and the elocity of the air. pe = elocity of plane relatie to earth pa = elocity of plane relatie to air ae = elocity of air relatie to earth 6

7 Relatie Velocity When flying with a crosswind, the plane s ground speed is the resultant of the elocity of the plane and the elocity of the air. pe = elocity of plane relatie to earth pa = elocity of plane relatie to air ae = elocity of air relatie to earth Pilots must fly with crosswind but not be sent off course. relatie elocity Sometimes the ector sums are more complicated! 7