KinematicsThe Study of Motion

Chapter 2

What are some different types of motion?

What are some terms (concepts) that describe our observations of motion?

Position

What questions do they answer?

Where?

Distance How far?

Time How long? When?Speed How fast?Acceleration How is the speed changing?

How do we operationally define the terms (concepts) that describe our observations of motion?

PositionMeasured in terms of distances (coordinates) from the axes of a one, two, or three dimensional coordinate system.

d1

d2

d3

DistanceChange in positionMeasured as the total length of the line segments and/or curves that trace the path of the object’s motion.

d1

d2

d3

distance = d1 + d2 + d3

Units: meters, m

Initial PositionFinal Position

DisplacementThe NET change in the position of an object.The length of the line segment joining the initial and final positions.

Independent of the path followed.

Units: meters, m

d1

d2

d3

distance = d1 + d2 + d3

Initial PositionFinal Position

Displacement

SpeedThe rate of change of position.Defined as the distance traveled divided by the time of travel.Note: Because speed can change during an object’s motion we will actually define average speed as the distance traveled divided by the time of travel.

average speed=distancetime Units: m

sSpeed may be:

Zero...

Constant but not zero...

Changing...

no motion

object travels equal distances in equal time intervals

object travels different distances in equal time intervals

Scalars and VectorsA scalar is any measured quantity having only a magnitude.

A vector is any measured quantity having both a magnitude AND an associated direction.

Examples:MassTimeTemperature

Example:Displacement

For motion in only one direction the direction can be specified with a + or - sign.Horizontal Motion: Toward the right (+) Toward the left (-)

Vertical Motion: Up (+) Down (-)

SpeedDistance

+/- Signs in Various ContextsUsed in front of a number:

+20 Number is greater than zero-20 Number is less than zero

Used in front of a vector:d = +20m Displacement is 20m to the right (or up)

d = -20m Displacement is 20m to the left (or down)

Between two numbers or vectors:

20m+10m Sum of two displacements 20m+10m = 30m

20m-(-10m) Difference between two displacements20m+(-10m) = 30m

Indicates magnitude

Indicates direction

Indicates an operation

+5m -5m

+5m -5m

Vector Example: DisplacementHorizontal Motion

Vertical Motion

Other VectorsSpeed is a scalar having only a magnitude.

The corresponding vector is called velocity.

average speed=distancetime

average velocity=displacementtime

+5m in 2s-5m in 2saverage velocity =−2.5 m

s

average velocity =+2.5 ms

velocity

AccelerationThe rate of change in the velocity

acceleration=change in velocitytime

Acceleration is a vector.

Units=ms2

Acceleration may be:

Zero…

Constant but not zero...

velocity constant.

Velocity changing uniformly, by the same amount each unit of time.

Changing... Velocity changing not uniformly, by the different amounts each unit of time.

Note: In this course we will not consider motion with a changing acceleration.

Acceleration can change the velocity in three ways.

If the acceleration is in the same direction as the velocity i.e., parallel to the velocity...

The magnitude of the velocity increases i.e., the object moves faster.Pressing the gas pedal in a car does this.

If the acceleration is in the opposite direction as the velocity i.e., antiparallel to the velocity...

The magnitude of the velocity decreases i.e., the object moves slower.

Pressing the brake pedal in a car does this.

If the acceleration is perpendicular the velocity...

The direction of the velocity changes.

Turning the steering wheel of a car does this,

A

B

C

Consider an object projected upward.

Because of gravity the object moves upward while at the same time going slower and slower.

It momentarily stops at point “B”.

It then moves downward while at the same time going faster and faster.

From point “A” to point “B” :Displacement is +

Velocity is + ; decreasingAcceleration is -, constant

At point “B” :Velocity is zero

Acceleration is -, constant

From point “B” to point “C” :Displacement is -

Velocity is - ; increasingAcceleration is -, constant

Summary of ConceptsConcept Type Operational Definition Units

Position Vector Measurements relative to varies 3-dimensional frame of reference.

Distance Scalar Length of path from initial meters, m position to final position.

Displacement Vector Length of line joining meters, m initial and final positions. (independent of path)

Time Scalar Measured relative to some seconds, s periodic phenomena.

Summary of Conceptscontinued

Concept Type Operational Definition Units

Speed Scalar Rate of change of position. m/s Distance divided by time. Magnitude of velocity.

Velocity Vector Rate of change of position. m/s Displacement divided by time.

Acceleration Vector Rate of change of velocity. m/s2

Change in velocity divided by time.

Symbolic-Mathematical Description of MotionSymbols:time→ tdisplacement→ dvelocity→ v (only used if the velocity is constant)average velocity→ v intial velocity→ vi

final velocity→ vf

acceleration→ achange in→ Δ→ final value−initial value

Definitions:

average velocity=displacementtime

v =dt

acceleration=change in velocitytime

a=Δvt =vf −vit

We will investigate and describe two types of motion:

Constant Velocity (acceleration = 0)

Constant, nonzero, Acceleration

Constant Velocity (acceleration = 0)Variables:

displacement, d

velocity, v…constant

time, t

Relationships:

v =dt

d =v⋅t

t =dv

(from definition)

a=0

Constant, nonzero, AccelerationVariables:

displacement, daverage velocity, v

initial velocity, vi

final velocity, vf

acceleration, atime, t

Relationships:

v =dt (from definition)

v =vi +vf2

d = v ⋅t

t = dv

a =vf −vi

t (from definition)

vf =vi + a⋅t

Only if acceleration is constant

Other Derived Relationships for Uniformly Accelerated Motion

Start with:

d = v ⋅tSubstitute:

v =vi +vf2 → d =

vi +vf2 ⋅t

Substitute:

vf =vi + a⋅t → d =vi + vi +a⋅t( )

2 ⋅t

Simplify: → d =2vi +a⋅t

2 ⋅t → d =(vi +12 a⋅t)⋅t

→ d =vi ⋅t+12 a⋅t2

Start With:

d =v t

Substitute:

v =vi +vf2 → d =

vi +vf2 ⋅t

Substitute

t =vf −via → d =

vf +vi2 ⋅

vf −via

Simplify:

d =(vf +vi)⋅(vf −vi)

2 ⋅a → d =vf2 −vi

2

2a

2a⋅d =vf2 −vi

2→ → vf

2 =vi2 + 2ad

SummaryConstant Velocity

Variables:

d

v

t

Relationships:

v =dt

d =v⋅t

t =dv

Constant AccelerationVariables: Relationships:

v =dt

v =vi +vf2

d = v ⋅t

t = dv

a =vf −vi

tvf =vi + a⋅t

d =vi ⋅t+12 a⋅t2

vf2 =vi

2 + 2ad

dv v i

v f

at

a=0

Free-FallFree-Fall includes all motion which meets the following two conditions:

1. Motion is only in the vertical direction (up/down)2. Motion is only affected by gravity

What type of motion is free-fall?

Constant VelocityConstant Acceleration

Does the acceleration depend on the mass of the object?Yes

No

What is the value of the acceleration due to gravity? Does it depend on the body producing the gravitational force? At the earth' s surface g = -9.8 m

s2

At the moon's surface g = -1.7 ms2

ConclusionFree-Fall is an example uniformly accelerated motion.At the earth’s surface the acceleration due to gravity is constant and independent of the object’s mass.

At the earth' s surface

the acceleration due to gravity, g = -9.8 ms2

The acceleration due to gravity depends on the mass and size of the body producing the gravitational force.

Does the acceleration depend on the direction (up/down)?

Yes

No