scalars vs. vectors scalar – a quantity that has a magnitude (size) but does not have a direction....
DESCRIPTION
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 mTRANSCRIPT
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