Scalars vs. Vectors Scalar – a quantity that has a magnitude (size) but does not have a direction....
-
Upload
jasper-bradley -
Category
Documents
-
view
220 -
download
0
description
Transcript of Scalars vs. Vectors Scalar – a quantity that has a magnitude (size) but does not have a direction....
![Page 1: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/1.jpg)
![Page 2: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/2.jpg)
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)
![Page 3: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/3.jpg)
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
![Page 4: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/4.jpg)
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
![Page 5: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/5.jpg)
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.
![Page 6: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/6.jpg)
Distance vs. DisplacementDistance vs. Displacement
A
BPath I
10 m
5 m
Total distance = 15mDisplacement = 11.1 m
Path II
![Page 7: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/7.jpg)
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.
![Page 8: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/8.jpg)
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
![Page 9: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/9.jpg)
![Page 10: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/10.jpg)
Traveling at a constant speed in a positive direction
position-time
velocity-time
acceleration-time
![Page 11: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/11.jpg)
Traveling at a constant speed in a negative direction
position-time
velocity-time
acceleration-time
![Page 12: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/12.jpg)
Remaining at rest
position-time
velocity-time
acceleration-time
![Page 13: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/13.jpg)
Gaining speed in a positive direction
position-time
velocity-time
acceleration-time
![Page 14: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/14.jpg)
Losing speed in a positive direction
position-time
velocity-time
acceleration-time
![Page 15: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/15.jpg)
Gaining speed in a negative direction
position-time
velocity-time
acceleration-time
![Page 16: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/16.jpg)
Losing speed in a negative direction
position-time
velocity-time
acceleration-time
![Page 17: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/17.jpg)
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
![Page 18: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/18.jpg)
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
![Page 19: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/19.jpg)
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
![Page 20: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/20.jpg)
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
![Page 21: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/21.jpg)
Graphical RelationshipsGraphical Relationships
SLOPESAREAS
![Page 22: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/22.jpg)
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
![Page 23: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/23.jpg)
Position to velocity vs. time Position to velocity vs. time graphgraph
![Page 24: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/24.jpg)
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
![Page 25: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/25.jpg)
Velocity to Position vs. Time Velocity to Position vs. Time GraphsGraphs
![Page 26: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/26.jpg)
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
![Page 27: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/27.jpg)
Instantaneous velocity from a Instantaneous velocity from a position vs. time graphposition vs. time graph
Slope of line tangent to curve at
specific time.
![Page 28: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/28.jpg)
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.
![Page 29: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/29.jpg)
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.
![Page 30: 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.](https://reader036.fdocuments.net/reader036/viewer/2022062401/5a4d1b147f8b9ab059990c0b/html5/thumbnails/30.jpg)
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