Unit 2 1-Dimensional Kinematics Kinematics --the science of describing the motion of objects Methods --using words, diagrams, numbers, graphs, and equations Terms -- scalars, vectors, distance, displacement, speed, velocity and acceleration

I Vector Analysis Objectives: What is vector quantity How to identify a physics quantity as scalar or vector type. How to graph vector How to add or subtract vectors in 1 dimensional space

velocity, acceleration, force, displacement 1 Scalars and Vectors Definition A scalar quantity is one that can be described by a number with proper unit (magnitude): Temperature (such as-5℃) Speed (such as 5m/s) Length (height as 10m) mass ( such as 2 lb, or 2kg) energy and power ( joule, watts) density(1kg/m3) time, time duration, distance A vector quantity is a quantity that is fully described by both magnitude and direction. velocity, acceleration, force, displacement

Distance moved is the actual length of the path along which an object travelled. It is a scalar. Displacement is the straight line distance from the beginning to the end of the path along which an object moved. It is a vector and has nothing to do with how the object got there.

2. About Vector: Vector quantity demands that both a magnitude and a direction are listed. Example: If say, "A bag of gold is outside the classroom. To find it, displace yourself 20 meters."  --not enough information to find the gold.  If say,  "A bag of gold is located outside the classroom. To find it, displace yourself from the center of the classroom door 20 meters in the west direction.“ –you will find it.

Examples of Vector Quantities: I travel 30 km in a Northerly direction (magnitude is 30 km, direction is North - this is a displacement vector) The train is going 80 km/h towards Sydney (magnitude is80 km/h, direction is 'towards Sydney' - it is a velocity vector) The force on the bridge is 50 N acting downwards (the magnitude is 50 Newtons and the direction is down - it is a force vector) Each of the examples above involves magnitude and direction

Check vector and scalars: http://www.physicsclassroom.com/class/1DKin/Lesson-1/Scalars-and-Vectors

3. Vector Notation Some book use a bold capital letter to name vectors. (example, a force vector F). Some textbooks write vectors using an arrow above the vector name, like this: 𝐹 In Graph: A vector is drawn using an arrow. The length of the arrow indicates the magnitude of the vector. The direction of the vector is represented by (not surprisingly :-) the direction of the arrow. 𝐴 𝐴 Not location related

Example 1 - Vectors vector A has direction 'up' and a magnitude of 4 cm. Vector B has the same direction as A, and has half the magnitude (2 cm). Vector C has the same magnitude as A (4 units), but it has different direction. Vector D is equivalent to vector A. It has the same magnitude and the same direction. It doesn't matter that A is in a different position to D - they are still considered to be equivalent vectors because they have the same magnitude and same direction. We can write: A = D Note: We cannot write A = C because even though A and C have the same magnitude (4 cm), they have different direction. They are not equivalent.

A zero vector has magnitude of 0. It can have any direction. This is called scaled vector diagram. Scale: 1cm=1km 4 A zero vector has magnitude of 0. It can have any direction.

4. VECTOR DIAGRAMS Vector diagrams are diagrams, which describe the direction and relative magnitude of a vector quantity by a vector arrow. For example, the velocity of a car moving down the road vector diagram: (Vector diagrams describe the velocity of a moving object during its motion)

5. Multiply vector by a scalar Multiply the magnitude, direction does not change for positive scalar, opposite for negative scalar 𝐴 4 𝐴 −3 𝐴

5. Vector Addition and Subtraction in 1 dimension A) One dimension vector addition. 𝐴 + 𝐵 = 𝑅 𝑅 is called resultant vector.  head-to-tail method  B) Subtraction: − 𝐵 𝐴 − 𝐵 = 𝐴 +(− 𝐵 )= 𝑅 𝑅 finish − 𝐵 start 𝐴

Follow Commutative Property of Addition Check page : http://www.intmath.com/vectors/2-vector-addition-1d.php 8 m 3 m 5 m 3 m +4 m -3 m 4 m 2 m +2 m 3 m 9 m In one dimension, we can use positive and negative numbers to represent vector: 𝐴 =+4m, 𝐵 =+2m, 𝐶 =−3m, then 𝑅 = 𝐴 + 𝐵 + 𝐶 =4m +2m+(-3m)=3m

Example 2: +5 + 5=+10 +5 +(- 5)=0 +5 + 10=+15 +5 +(-10) = -5 5 – 15 = -10 10 +(-5) = 5

6. Vectors in 2 dimensional space A vector in 2 dimensional space usually is represented by its magnitude and direction angle: vector 𝐴 (2 km, 30o ) Note: the angle is CCW from positive x-axis, here is from east direction.

7. Adding Vectors in 2 dimensional space  head-to-tail method 

Example 3 Scale: 1cm=1m 𝐵 𝑅 2.00 m 𝐴 6.00 m

Example 2 continue 𝑅 2.00 m 6.00 m

Adding more than 2 vectors If displacement vectors A, B, and C are added together, the result will be vector R. As shown in the diagram, vector R can be determined by the use of an accurately drawn, scaled, vector addition diagram. example

Example 4. vector walk Either using centimeter-sized displacements upon a map or meter-sized displacements in a large open area, a student makes several consecutive displacements beginning from a designated starting position.  Draw the resultant from the tail of the first vector to the head of the last vector 𝑅

Play the simulation in site: https://phet. colorado