1. AP Physics C Mrs. Coyle

2. Coordinate Systems Vectors and Scalars Properties of Vectors Unit Vectors

3. Coordinate Systems Cartesian (rectangular) Coordinates (x,y) Polar Coordinates (r, θ) –θ is taken to be positive counterclockwise from the +x axis.

4. Vectors and Scalars Scalars- magnitude only Vectors- magnitude and direction

5. Equality of Vectors Two vectors are equal if they have the same magnitude and direction.

6. Addition of Vectors Graphical Algebraic Resultant: sum of vectors

7. Properties of Vector Addition –Commutative Property of Addition A + B = B + A –Associative Property of Addition (A + B) + C = A + (B + C)

8. Graphical Addition of Vectors Head-to-Tail Method Parallelogram Method

9. Graphical Addition of Vectors Head-to-Tail Method Vectors are moved parallel to themselves so that they are positioned in such a way that the head of one is adjacent to the tail of the other. The resultant is drawn by starting at the first tail (loose tail) and ending (arrow head pointed) at the last head (loose head). A B Resultant

10. Graphical Addition of Vectors Parallelogram Method The vectors are placed tail to tail forming a rectangle. The diagonal that starts at the joint tails has its tail at the joint tails) is the resultant. A B Resultant

11. Graphical Vector Subtraction When subtracting A-B : Invert vector B to get -B Add A+(-B) normally

12. Algebraic Addition of Vectors- Component Method 1)Find x and y components of each vector. a x = acosθ a y = a sinθ

13. Component Method Cont’d 2)Add x and y components. 3)Use the Pythagorean Theorem to find the magnitude of the resultant. 4)Use =tan -1 |Y | to find the direction X with respect to the x-axis.

14. Unit Vectors: î, ĵ, k Dimensionless vector with a magnitude of 1. They specify direction x, y, z Example: A= 2 î + 3 ĵ - 6k

15. Example 1 Add the vectors: A= 10 î - 1 ĵ - 6k B= - 6î + 5 ĵ + 6k Give the components of the resultant vector, its magnitude and its direction with respect to the x- axis. Answer: R= 4î + 4ĵ, 5.7, 45 deg above +x axis

16. Example 2 The position vector as a function of time for an object is given by r(t)= 2 î + 3t ĵ - 6k, r is in meters and t is in seconds. Evaluate dr/dt and explain what is its significance?

17. Example 3 These are instructions for finding a treasure : Go 75.0 paces at 240°, turn to 135° and walk 125 paces, then travel 100 paces at 160°. The angles are measured ccw from the east, the +x direction. Determine the resultant displacement from the starting point. Answer: 227 paces at 165°

18. Useful Link http://hyperphysics.phy- astr.gsu.edu/hbase/vect.html#vecco nhttp://hyperphysics.phy- astr.gsu.edu/hbase/vect.html#vecco n