Scalars vs. VectorsScalars vs. VectorsScalarScalar – a quantity that has a – a quantity that has a

magnitude (size) but does not have a magnitude (size) but does not have a direction. Ex. # of objects (5 apples) , direction. Ex. # of objects (5 apples) , speed (10 m.p.h.), distance (34 miles)speed (10 m.p.h.), distance (34 miles)

VectorVector – a quantity that has – a quantity that has magnitude (size) magnitude (size) ANDAND has a has a direction. Ex. Velocity (10 m.p.h. @ direction. Ex. Velocity (10 m.p.h. @ 454500), displacement (34 miles NE)), displacement (34 miles NE)

VectorsVectorsVectors are represented by an arrowVectors are represented by an arrowThe length of the arrow is The length of the arrow is

proportional to the magnitude of the proportional to the magnitude of the vector it representsvector it represents

10 m 20 m

VectorsVectorsIn 1-D the direction of the arrow is In 1-D the direction of the arrow is

indicated by terms like left or right, up indicated by terms like left or right, up or down. Typically the direction is or down. Typically the direction is defined by defined by signssigns, either a plus (+) or a , either a plus (+) or a minus (-).minus (-).

When working on a problem define your When working on a problem define your directions signs firstdirections signs first

You know a quantity is a vector if it is You know a quantity is a vector if it is boldface or has an arrow over itboldface or has an arrow over it

Ex. Ex. vv or v or v

Position, Displacement, and Position, Displacement, and DistanceDistance

Position (Position (pospos)) – where an object is – where an object is located on a number line. Ex. 25 meter located on a number line. Ex. 25 meter mark.mark.

Distance (Distance (dd)) – the total path length – the total path length traveled to get from one position to traveled to get from one position to another.another.

Displacement (Displacement (ΔΔd)d) – the distance – the distance from start to finish no matter how the from start to finish no matter how the object traveled between the two points. object traveled between the two points. The length of the change of position.The length of the change of position.

Distance vs. DisplacementDistance vs. Displacement

A

BPath I

10 m

5 m

Total distance = 15mDisplacement = 11.1 m

Path II

Some Other Motion Some Other Motion DefinitionsDefinitions

VelocityVelocity – the rate at which a – the rate at which a displacement is covered. This is a displacement is covered. This is a vector quantity.vector quantity.

SpeedSpeed – the rate at which a distance – the rate at which a distance is covered. This is a scalar quantity.is covered. This is a scalar quantity.

v = Δd/Δt“average” velocity

m/sHow fast the position is changing. How fast the

object is moving.

Some Other Motion Some Other Motion DefinitionsDefinitions

Acceleration (a)Acceleration (a) – the rate at which – the rate at which the velocity changes. This is a vector the velocity changes. This is a vector quantity.quantity.

a = Δv/ΔtHow fast the

velocity is changing

m/s/sm/s2

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 What you can tell from looking at a position vs. time graphat a position vs. time graph

Above x-axis = positive positionAbove x-axis = positive position Below x-axis = negative positionBelow x-axis = negative position Positive Slope = positive velocity (direction)Positive Slope = positive velocity (direction) Negative Slope = negative velocity Negative Slope = negative velocity

(direction)(direction) Zero Slope = at restZero Slope = at rest Linear = constant velocityLinear = constant velocity Increasing steepness = speeding upIncreasing steepness = speeding up Decreasing steepness = slowing downDecreasing steepness = slowing down

What you can tell from looking What you can tell from looking at a velocity vs. time graphat a velocity vs. time graph

Above x-axis = positive velocity (direction)Above x-axis = positive velocity (direction) Below x-axis = negative velocity Below x-axis = negative velocity

(direction)(direction) Positive Slope = positive acceleration Positive Slope = positive acceleration Negative Slope = negative accelerationNegative Slope = negative acceleration Zero Slope = no acceleration (constant Zero Slope = no acceleration (constant

speed)speed) Increase in # Value = Speeding upIncrease in # Value = Speeding up Decrease in # Value = Slowing downDecrease in # Value = Slowing down

What can you tell from looking What can you tell from looking at an acceleration vs. time graphat an acceleration vs. time graphAbove x-axis = positive accelerationAbove x-axis = positive accelerationBelow x-axis = negative accelerationBelow x-axis = negative acceleration

Graphical Indicators of MotionGraphical Indicators of MotionPosition vs.

TimeVelocity vs.

TimeAcceleration vs.

TimeInstantaneous

PositionValue on y-

axisN/A 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 Velocity

N/A Change in y value

Area from graph to x-axis

Instantaneous Acceleration

N/A Slope of tangent line at specific

time

Value on y-axis

Graphical RelationshipsGraphical Relationships

SLOPESAREAS

Position to velocity-time Position to velocity-time graphgraph

Slopes = VelocityA-B: slope = (10-10)/(2-0) slope = 0 m/sB-C: slope = (25-10)/(5-2) slope = 5 m/sC-D: slope = (25-25)/(6-5) slope = 0 m/sD-E: slope = (-5-25)/(9-6) slope = -10 m/sE-F: slope = (-8-(-5)/(12-9) slope = -1 m/s

Position to velocity vs. time Position to velocity vs. time graphgraph

Velocity to position vs. time Velocity to position vs. time graphsgraphs AREA =

Displacement1-3 s: A = 2m/s x 3s = 6 m3-7 s: A = 3m/s x 4 s = 12 m7-9 s: A = 1m/s x 2s = 2 m9-10 s: A = 0m/s x 1s = 0 m10-15 s: A = -4 m/s x 5s = -20 m

Velocity to Position vs. Time Velocity to Position vs. Time GraphsGraphs

Velocity to acceleration vs. time Velocity to acceleration vs. time graphs and vice versagraphs and vice versa

Use the same procedure for these Use the same procedure for these graphs as you did for the position graphs as you did for the position and velocity vs. time graphsand velocity vs. time graphs

Instantaneous velocity from a Instantaneous velocity from a position vs. time graphposition vs. time graph

Slope of line tangent to curve at

specific time.

Instantaneous acceleration from Instantaneous acceleration from a velocity-time grapha velocity-time graph

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

Kinematic EquationsKinematic EquationsThese equations are derived by These equations are derived by

taking the slope and area of a taking the slope and area of a velocity vs. time graph.velocity vs. time graph.

These equations are only valid for an These equations are only valid for an object with a object with a constantconstant acceleration. acceleration.

Kinematic EquationsKinematic EquationsBIG 3BIG 3ΔΔd = vd = viit + ½ att + ½ at22

vvff = v = vii + at + atvvff

22 = v = vii22 + 2a( + 2a(ΔΔd)d)

OTHEROTHERΔΔd = ½ (vd = ½ (vii + v + vff)t)t