Chapter 3: Motion in a Plane Vector Addition Velocity Acceleration Projectile motion Relative Velocity CQ: 1, 2. P: 3, 5, 7, 13, 21, 31, 39, 49, 51.
Two Dimensional Vectors Displacement, velocity, and acceleration each have (x, y) components Two methods used: geometrical (graphical) method algebraic (analytical) method / 2
Graphical, Tail-to-Head 3
Addition Example Giam (11)
Order Independent (Commutative) 5
Subtraction, tail-to-tail 6
Subtraction Example Giam (19)
Algebraic Component Addition trigonometry & geometry “R” denotes “resultant” sum R x = sum of x-parts of each vector R y = sum of y-parts of each vector 8
Vector Components 9
Examples Magnitude || (g4-5) Notation, ExampleMagnitude || (g4-5) Component Example AnimatedComponent Example Phet Vectors 10
Trigonometry o a h 11
Using your Calculator: Degrees and Radians Check this to verify your calculator is working with degrees 12
Example: o a h Given: = 10°, h = 3 Find o and a. 13
Inverse Trig Determine angle from length ratios. Ex. o/h = 0.5: Ex. o/a = 1.0: 14
Pythagorean Theorem o a h Example: Given, o = 2 and a = 3 Find h 15
Azimuth: Angle measured counter- clockwise from +x direction. Examples: East 0°, North 90°, West 180°, South 270°. Northeast = NE = 45° 16
Check your understanding: Note: All angles measured from east. 17 What are the Azimuth angles? A: B: C: 180° 60° 110° 70°
Components: Given A = 25°, its x, y components are: Check using Pythagorean Theorem: 18
Vector Addition by Components: 19
R = (10cm, 0°) + (10cm, 45°): 20 Example Vector Addition
(cont) Magnitude, Angle: 21
General Properties of Vectors size and direction define a vector location independent change size and/or direction when multiplied by a constant Vector multiplied by a negative number changes to a direction opposite of its original direction. written: Bold or Arrow 22
these vectors are all the same 23
Multiplication by Constants A -A 0.5A -1.2A 24
Projectile Motion time = 0: e.g. baseball leaves fingertips time = t: e.g. baseball hits glove Horizontal acceleration = 0 Vertical acceleration = -9.8m/s/s Horizontal Displacement (Range) = x Vertical Displacement = y V o = launch speed o = launch angle
Range vs. Angle 26
Example 1: 6m/s at 30 v o = 6.00m/s o = 30° x o = 0, y o = 1.6m; x = R, y = 0 27
Example 1 (cont.) Step 1 28
Quadratic Equation 29
Example 1 (cont.) End of Step 1 30
Example 1 (cont.) Step 2 (a x = 0) “Range” = 4.96m End of Example 31
Relative Motion Examples: people-mover at airport airplane flying in wind passing velocity (difference in velocities) notation used: velocity “BA” = velocity of B – velocity of A 32
Summary Vector Components & Addition using trig Graphical Vector Addition & Azimuths Projectile Motion Relative Motion 33
R = (2.0m, 25°) + (3.0m, 50°): 34
(cont) Magnitude, Angle: 35
PM Example 2: v o = 6.00m/s o = 0° x o = 0, y o = 1.6m; x = R, y = 0 36
PM Example 2 (cont.) Step 1 37
PM Example 2 (cont.) Step 2 (a x = 0) “Range” = 3.43m End of Step 2 38
PM Example 2: Speed at Impact 39
v1 1. v1 and v2 are located on trajectory. a 40
Q1. Givenlocate these on the trajectory and form v. 41
Kinematic Equations in Two Dimensions * many books assume that x o and y o are both zero. 42
Velocity in Two Dimensions v avg // r instantaneous “v” is limit of “v avg ” as t 0 43
Acceleration in Two Dimensions a avg // v instantaneous “a” is limit of “a avg ” as t 0 44
Conventions r o = “initial” position at t = 0 r = “final” position at time t. 45
Displacement in Two Dimensions roro r rr 46
Acceleration ~ v change 1 dim. example: car starting, stopping 47
Acceleration, Dv, in Two Dimensions 48
Ex. Vector Addition Add A = azimuth, plus B = azimuth. Find length of A+B, and its azimuth. Sketch the situation.
Ex.2: + Length and azimuth?
Calculate F3 F1 = F2 = F1 + F2 + F3 = 0 F3 = -(F1 + F2) Rx = 8cos60+5.5cos(-45)=7.89 Ry = 8sin60+5.5sin(-45)=3.04 R = 8.46 Angle = tan-1(3.04/7.89) = 21 deg above +x axis Answer book wants is 180 off this angle!
Addition by Parts (Components) 52