Scalars vs. Vectors Scalar – a quantity that has a magnitude (size) but does not have a direction. Ex. # of objects (5 apples), speed (10 m.p.h.), distance (34 miles) Scalar – a quantity that has a magnitude (size) but does not have a direction. Ex. # of objects (5 apples), speed (10 m.p.h.), distance (34 miles) Vector – a quantity that has magnitude (size) AND has a direction. Ex. Velocity ( ), displacement (34 miles NE) Vector – a quantity that has magnitude (size) AND has a direction. Ex. Velocity ( ), displacement (34 miles NE)

Vectors Vectors are represented by an arrow Vectors are represented by an arrow The length of the arrow is proportional to the magnitude of the vector it represents The length of the arrow is proportional to the magnitude of the vector it represents 10 m20 m

Vectors In 1-D the direction of the arrow is indicated by terms like left or right, up or down. Typically the direction is defined by signs, either a plus (+) or a minus (-). In 1-D the direction of the arrow is indicated by terms like left or right, up or down. Typically the direction is defined by signs, either a plus (+) or a minus (-). When working on a problem define your directions signs first When working on a problem define your directions signs first You know a quantity is a vector if it is boldface or has an arrow over it You know a quantity is a vector if it is boldface or has an arrow over it Ex. v or v Ex. v or v

Position, Displacement, and Distance Position (pos) – where an object is located on a number line. Ex. 25 meter mark. Position (pos) – where an object is located on a number line. Ex. 25 meter mark. Distance (d) – the total path length traveled to get from one position to another. Distance (d) – the total path length traveled to get from one position to another. Displacement (Δd) – the distance from start to finish no matter how the object traveled between the two points. The length of the change of position. Displacement (Δd) – the distance from start to finish no matter how the object traveled between the two points. The length of the change of position.

Distance vs. Displacement A B Path I 10 m 5 m Total distance = 15m Displacement = 11.1 m Path II

Some Other Motion Definitions Velocity – the rate at which a displacement is covered. This is a vector quantity. Velocity – the rate at which a displacement is covered. This is a vector quantity. Speed – the rate at which a distance is covered. This is a scalar quantity. Speed – the rate at which a distance is covered. This is a scalar quantity. v = Δd/Δt “average” velocity m/s How fast the position is changing. How fast the object is moving.

Some Other Motion Definitions Acceleration (a) – the rate at which the velocity changes. This is a vector quantity. Acceleration (a) – the rate at which the velocity changes. This is a vector quantity. a = Δv/Δt How fast the velocity is changing m/s/s m/s 2

Traveling at a constant speed in a positive direction position- time velocity- time acceleration -time

Traveling at a constant speed in a negative direction position- time velocity- time acceleration -time

Remaining at rest position- time velocity- time acceleration -time

Gaining speed in a positive direction position- time velocity- time acceleration -time

Losing speed in a positive direction position- time velocity- time acceleration -time

Gaining speed in a negative direction position- time velocity- time acceleration -time

Losing speed in a negative direction position- time velocity- time acceleration -time

What you can tell from looking at a position vs. time graph Above x-axis = positive position Above x-axis = positive position Below x-axis = negative position Below x-axis = negative position Positive Slope = positive velocity (direction) Positive Slope = positive velocity (direction) Negative Slope = negative velocity (direction) Negative Slope = negative velocity (direction) Zero Slope = at rest Zero Slope = at rest Linear = constant velocity Linear = constant velocity Increasing steepness = speeding up Increasing steepness = speeding up Decreasing steepness = slowing down Decreasing steepness = slowing down

What you can tell from looking at a velocity vs. time graph Above x-axis = positive velocity (direction) Above x-axis = positive velocity (direction) Below x-axis = negative velocity (direction) Below x-axis = negative velocity (direction) Positive Slope = positive acceleration Positive Slope = positive acceleration Negative Slope = negative acceleration Negative Slope = negative acceleration Zero Slope = no acceleration (constant speed) Zero Slope = no acceleration (constant speed) Increase in # Value = Speeding up Increase in # Value = Speeding up Decrease in # Value = Slowing down Decrease in # Value = Slowing down

What can you tell from looking at an acceleration vs. time graph Above x-axis = positive acceleration Above x-axis = positive acceleration Below x-axis = negative acceleration Below x-axis = negative acceleration

Graphical Indicators of Motion Position vs. Time Velocity vs. Time Acceleration vs. Time Instantaneous Position Value on y- axis N/A Displacement Change in y value Area from graph to x- axis N/A Instantaneous Velocity Slope of tangent line at specific time Value on y- axis N/A Change in VelocityN/A Change in y value Area from graph to x-axis Instantaneous Acceleration N/ASlope of tangent line at specific time Value on y-axis

Graphical Relationships SLOPES AREAS

Position to velocity-time graph Slopes = Velocity A-B: slope = (10-10)/(2-0) slope = 0 m/s B-C: slope = (25-10)/(5-2) slope = 5 m/s C-D: slope = (25-25)/(6-5) slope = 0 m/s D-E: slope = (-5-25)/(9-6) slope = -10 m/s E-F: slope = (-8-(-5)/(12-9) slope = -1 m/s

Position to velocity vs. time graph

Velocity to position vs. time graphs AREA = Displacement 1-3 s: A = 2m/s x 3s = 6 m 3-7 s: A = 3m/s x 4 s = 12 m 7-9 s: A = 1m/s x 2s = 2 m 9-10 s: A = 0m/s x 1s = 0 m s: A = -4 m/s x 5s = -20 m

Velocity to Position vs. Time Graphs

Velocity to acceleration vs. time graphs and vice versa Use the same procedure for these graphs as you did for the position and velocity vs. time graphs Use the same procedure for these graphs as you did for the position and velocity vs. time graphs

Instantaneous velocity from a position vs. time graph Slope of line tangent to curve at specific time.

Instantaneous acceleration from a velocity-time graph Same procedure as getting instantaneous velocity from a position vs. time graph. You take the slope of a tangent line drawn at a specific time. Same procedure as getting instantaneous velocity from a position vs. time graph. You take the slope of a tangent line drawn at a specific time.

Kinematic Equations These equations are derived by taking the slope and area of a velocity vs. time graph. These equations are derived by taking the slope and area of a velocity vs. time graph. These equations are only valid for an object with a constant acceleration. These equations are only valid for an object with a constant acceleration.

Kinematic Equations BIG 3 BIG 3 Δd = v i t + ½ at 2 Δd = v i t + ½ at 2 v f = v i + at v f = v i + at v f 2 = v i 2 + 2a(Δd) v f 2 = v i 2 + 2a(Δd) OTHER OTHER Δd = ½ (v i + v f )t Δd = ½ (v i + v f )t