Chapter 3

I. Vector Addition
    A. Since vectors have direction, these directions need to be taken into account when the vectors are added.
    B. Graphical Methods
        1. Both use scale drawing
        2. Parallelogram Method
            a. Can add only two vectors at a time
            b. Common origin
            c. Example - diagram
        3. Polygon Method
            a. Can add any number of vectors at one time
            b. "Tip - to - tail;" the terminal end of one vector is the initial end of the next vector
            c. Example - diagram
    C. Mathematical Methods
        1. Triangle Method
            a. Can only add two vectors at a time
            b. Generally will need to use Law of Sines and Law of Cosines
            c. We will not use this method
        2. Component Method
            a. Can add any number of vectors at one time
            b. Break each vector into its components
            c. Add all the x-components and add all the y-components
            d. The sums of the components of the individual vectors give the components of the resultant
            e. Use the Pythagorean Theorem and the tangent to find the resultant
            f. Example

II. Projectile Motion
     A. Four rules to keep in mind
        1. The motion in the x-direction is independent of the motion in the y-direction.
        2. The motion in the x-direction is constant velocity (a = 0)
        3. The motion in the y-direction is free fall (a = g downward)
        4. The time is the same for both directions
    B. Two cases
        1. Projected horizontally
            a. Initial velocity is all in x-direction, v_yo = 0
            b. Example
        2. General projectile
            a. Use components of the initial velocity
            b. Example

1. Components of a vector

Think of a vector as a distance at some angle on the Cartesian co-ordinate system. We need to find the distance along the x- and the y-axes.

x - component: v_x = v cos y - component: v_y = v cos

v = 100.0 cm at 25.0°

v_x = v cos = (100.0 cm) cos 25.0° = 90.63 cm v_y = v cos = (100.0 cm) sin 25.0° = 42.46 cm

2. Full component method

Vector x-component: v_x = v cos y-component: v_y = v sin

10.0 m @ 55.0° 5.74 m 8.19 m

15.0m @ 128° -9.24 m 11.82 m

5.70 m @ 225° -4.03 m -4.03 m

10.0 m @ 110° -3.42 m 9.40 m

R -10.95 m 25.38

R = [R_x² + R_x²]^½ = [(-10.95 m)² + (25.38 m)²]^½ = 27.6 m

Reference Angle = tan^-1 (|R_y|/|R_x|) = tan^-1 (|25.38 m|/|-10.95 m|) = 66.7° in second quadrant (- x and + y component)

Standard position angle = 180° - reference angle = 113.3°

Vector	x-component: v_x = v cos	y-component: v_y = v sin
10.0 m @ 55.0°	5.74 m	8.19 m
15.0m @ 128°	-9.24 m	11.82 m
5.70 m @ 225°	-4.03 m	-4.03 m
10.0 m @ 110°	-3.42 m	9.40 m
R	-10.95 m	25.38