CHAPTERS 3 & Picturing Motion Motion Diagrams A series of consecutive frames (frame by frame) of the motion of an object. Similar to movie film (30 frames per second).

The Particle Model Motion diagram of wheel with two different dots, center of wheel and edge of wheel.

3.2 Where & When? Coordinate Systems 2 dimensional AKA x-y coordinate system

Vectors and Scalars Scalar Quantity= A quantity that tells you only the magnitude (size/amount) of something. A number with units. Examples: 42kg, 100 o C, 40s, 10hr, $89, 100m/s,

Vector Quantity Vector Quantity= Magnitude with direction. Example: 46km/hr North, 15m/s SW, 58km→ Vectors are drawn to scale, the larger the vector, the larger the magnitude. Examples on board.

Vectors

Time Intervals and Displacement Displacement (Δd) = The distance and direction between two positions. Δd = d f – d i d f = final position d i = initial position Δd can be positive/negative Examples on number line.

Time Interval (Δt) = The time required for an object to complete some displacement. Δt = t f – ti t f = final time t i = initial time (usually t i = 0)  Δt is always positive

3.3 Velocity and Acceleration Velocity Speed vs velocity, is there a difference or are the terms interchangeable? Speed examples: 57m/s, 37km/hr, 17cm/yr, 68mph  speed is a scalar quantity.

Velocity = Speed with direction, a vector quantity. Examples: 57ms East, 37km/hr SE, 17 cm/yr→, 68mph West 17 cm/yr→, 68mph West

300 o

38 o N of W

Average velocity _ Δd d f - d i _ Δd d f - d i Avg vel (v)= Δt = t f – t i ________ ________ avg vel = vector quantity (speed&direction) Examples on number line.

The frog jumps from 0m to 9m in 3 seconds, what is the frog’s avg vel?

Δd df – di Δd df – di avg vel = Δt = tf – ti = 9m – 0m 9m 3s – 0s = 3s = 3m/s

Other Examples 1. A runner begins at the starting line and crosses the 80m finish line in 4 seconds. What is the runner’s average velocity? A = 20m/s 2. A car travels from the school, 200km West in 5hr. What is the car’s average velocity? A = -40km/hr

Instantaneous Velocity = ? The speed and direction of an object at a particular “instant” in time, how fast it is moving right now. Examples of instantaneous velocity = Speedometer, radar gun, tachometer

Motion Diagram of Golf Ball Examples on Board: Golf putting right/left (+v, -v, )

EQUATIONS V = v i + atv = vel (inst/final) v i = initial velocity V 2 = v i 2 + 2a ΔdΔd = d f - d i d f = d i + vΔt

Acceleration (a) = The rate of change in velocity (Δv). Δv v f - v i Δv v f - v i a = Δt = t f - t i Example: A car pulls out from a stop sign, 20s later it is traveling West at 40m/s. What is the car’s acceleration.

Δv v f - v i -40m/s - 0m/s Δv v f - v i -40m/s - 0m/s a = Δt = t f - t i = 20s - 0s -40m/s -40m/s a = 20s = -2m/s 2

Examples of acceleration Motion Diagrams Car speeding up, then constant velocity, then slowing down.

All problems all year for every chapter MUST have the following or points will be deducted. HOMEWORK INCLUDED! 1. A sketch/drawing or FBD. 2. Table of Known/unknown values. 3. Write the equation(s) to be used. 4. Plug in numbers 5. Show ALL steps/work.

More Equations v f 2 = v i 2 + 2aΔd v f 2 = v i 2 + 2a(d f - d i ) v f 2 = v i 2 + 2a(d f - d i )

4.1 Properties of Vectors Graphical representation = An arrow is drawn to scale and at the proper direction. The length of the arrow represents the magnitude of the vector.

VECTOR EXAMPLES

Resultant Vector (R) = The sum of 2 or more vectors. R = vector 1 + vector 2 + vector 3 +…

Resultant Vector Example Examples on board 1. 2 equal vectors 2. 2 equal/opposite vectors 3. 2 vectors 90 o apart

Resultant Vector Example

From SVHS to home various examples. No matter which route you take the displacement (Δd) will be the same.

Graphical Addition of Vectors Vectors are drawn from tip to tail

Resultant Vector (R) = The sum of two or more vectors. The order of adding the vectors does not matter, just like adding any other values. Since the vectors are drawn to scale, the magnitude of the resultant ( R ) can be measured with a ruler.

Resultant Vector Example

Example on board: 2 different paths – use meter stick The resultant vectors (R), are equal, the path does not matter, when all the individual vectors are added together the resultants will be equal.

What is the magnitude of R for the vectors below?

What is the magnitude of the R vector below?

What is the magnitude of the vectors below? The red vector has a magnitude of 40, the purple vector 65, use the indicated angles.

Can you use the Pythagorean Theorem? NO! Why not? Because the triangle is NOT a right triangle. How can you solve for the resultant? You must use the LAW of Cosines.

Law of Cosines Equation R 2 = A 2 + B 2 - 2ABcos  R 2 = – 2(40)(65)(cos119 o ) R 2 = – 5200( ) R 2 = R 2 = R = √¯ R = 91.37

Relative Velocities: Some Applications You are in a bus traveling at a velocity of 8m/s East. You walk towards the front of the bus at 3m/s, what is your velocity relative to the street? 11m/s 11m/s

You are in a bus traveling at a velocity of 8m/s East. You walk towards the back of the bus at 3m/s, what is your velocity relative to the street? 5m/s

A plane is traveling North at 800km/h, the wind is blowing East at 150km/h. What is the speed of the plane relative to the ground? 1. Draw sketch of vectors 2. List known/unknown 3. Write equation 4. Plug in numbers 5. Solve showing all work/steps

Solve Example Use Pythagorean Theorem A 2 + B 2 = C 2 Vpg 2 = Vp 2 + Vw 2 Vpg = √¯Vp 2 + Vw 2 Vpg = √¯(800km/h) 2 + (150km/h) 2 Vpg = √¯640,000km 2 /h ,500km 2 /h 2 Vpg = √¯662,500km 2 /h 2 Vpg = 813.9km/h

Boat/River Example A river flows South at 8m/s, a boat travels due East at 15m/s. Where will the boat end up and what will the boat’s speed be relative to the shore? 1. Draw sketch of vectors 2. List known/unknown 3. Write equation 4. Plug in numbers 5. Solve showing all work/steps

C 2 = A 2 + B 2 R 2 = V B 2 + V R 2 V BS 2 = V B 2 + V R 2 V BS 2 = (15m/s) 2 + (8m/s) 2 V BS = √¯(225m 2 /s 2 ) + (64m 2 /s 2 ) V BS = √¯289m 2 /s 2 V BS = 17m/s

4.2 Components of Vectors Choosing a Coordinate System

Draw Coordinate system with a vector. The angle (  ) tells us the direction of the vector. The direction of the vector is defined as the angle that the vector measures counterclockwise from the positive x-axis.

Components The vector “A” can be resolved into two component vectors. A x = parallel to the x-axis A y = parallel to the y-axis A = A x + A y Vector Resolution = the process of breaking down a vector into its x & y components.

Components = the magnitude and sign of the component vectors. Algebraic calculations only involve the components of vectors not the vectors themselves. ADJ A x ADJ A x A x = Acos  cos  = HYP = A OPP A Y OPP A Y A Y = Asin  sin  = HYP = A

EXAMPLE PROBLEM-1 A car travels 72km on a straight road at 25 o. What are the x and y component vectors? A x = 65.25km A Y = 30.43km

Example Problem - 2 A bus travels 37km North, then 57km East. What is the displacement and direction of the resultant.

Example Problem - 3 A runner travels at 15m/s west, then 13m/s south. What is the magnitude and direction (expressed all three ways) of the runner’s velocity.

Algebraic Addition of Vectors opp opp sin  = hyp adj adj cos  = hyp opp opp tan  = adj

Two or more vectors (A, B, C, …) may be added by: 1. resolving each vector into its x & y components 2. add the x-components together to form R x = A x + B x + C x + … 3. add the y-components together to form R Y = A Y + B Y + C Y + … R = R x + R y

Add 3 vectors Example

Vector Addition Algebraically What is the magnitude and direction (expressed all three ways) of the following vectors? 1) 14km east4) 14km north 2) 10km east5) 8km south 3) 7km west6) 15km south