RENE DESCARTES (1736-1806)

1. Motion In Two Dimensions RENE DESCARTES (1736-1806)

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

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

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

5. 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 applet

6. 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! simple vector addition right triangle vector addition non-right triangle vector addition Vector Subtraction To subtract a vector simply add the opposite vector. click for applet click for applet simple vector subtraction right triangle vector subtraction

7. Resolving Vectors It is useful to resolve or “break down” vectors into component vectors.

8. Finding a Resultant Often a vector’s components are known, and the resultant of these components must be found. Finding the magnitude of the resultant: Finding the direction of the resultant:

9. 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. Vertical velocity: Horizontal velocity:

10. 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? Finding the rocket’s speed (magnitude) Finding the rocket’s angle (direction)

11. Projectile Motion vx vx vx vx vx vx vx vx vy vy vy v vy vy v vy vy vy v Horizontal: constant motion, ax = 0 Projectile motion = constant motion + freefall motion Vertical: freefall motion, ay = g = •9.8 m/s2 velocity is tangent to the path of motion resultant velocity: click for applet θ click for applet

12. Projectile Motion at an Angle vx vx vx vx vx v vy vy v vx vy v vx vy v v vy vyi vx vy v v v vy vx vx vy v vertical velocity, vy is zero here! velocity components: θ θ click for applet click for applet

13. 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. click for reference frame applet click for relative velocity applet refers to the object refers to the reference frame 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

14. 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. vpg= velocity of person relative to ground vps= velocity of person relative to sidewalk vsg= 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. vpe= velocity of plane relative to earth vpa= velocity of plane relative to air vae= velocity of air relative to earth

15. 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. vpe= velocity of plane relative to earth vpa= velocity of plane relative to air vae= velocity of air relative to earth Pilots must fly with crosswind but not be sent off course. Sometimes the vector sums are more complicated!