Chapter 2Chapter 2

Motion in One Motion in One DimensionDimension

DynamicsDynamics

The branch of physics involving the The branch of physics involving the motion of an object and the motion of an object and the relationship between that motion relationship between that motion and other physics conceptsand other physics concepts

KinematicsKinematics is a part of dynamics is a part of dynamics In kinematics, you are interested in In kinematics, you are interested in

the the descriptiondescription of motion of motion NotNot concerned with the cause of the concerned with the cause of the

motionmotion

Brief History of MotionBrief History of Motion

Sumaria and EgyptSumaria and Egypt Mainly motion of heavenly bodiesMainly motion of heavenly bodies

GreeksGreeks Also to understand the motion of Also to understand the motion of

heavenly bodiesheavenly bodies Systematic and detailed studiesSystematic and detailed studies

““Modern” Ideas of MotionModern” Ideas of Motion

GalileoGalileo Made astronomical observations with Made astronomical observations with

a telescopea telescope Experimental evidence for description Experimental evidence for description

of motionof motion Quantitative study of motionQuantitative study of motion

PositionPosition

Defined in terms Defined in terms of a of a frame of frame of referencereference One dimensional, One dimensional,

so generally the x- so generally the x- or y-axisor y-axis

Vector QuantitiesVector Quantities

Vector quantities need both Vector quantities need both magnitude (size) and direction to magnitude (size) and direction to completely describe themcompletely describe them Represented by an arrow, the length of Represented by an arrow, the length of

the arrow is proportional to the the arrow is proportional to the magnitude of the vectormagnitude of the vector

Head of the arrow represents the Head of the arrow represents the directiondirection

Generally printed in bold face typeGenerally printed in bold face type

Scalar QuantitiesScalar Quantities

Scalar quantities are completely Scalar quantities are completely described by magnitude onlydescribed by magnitude only

DisplacementDisplacement

Measures the change in position Measures the change in position Represented as Represented as x (if horizontal) or x (if horizontal) or y y

(if vertical)(if vertical) Vector quantityVector quantity

+ or - is generally sufficient to indicate + or - is generally sufficient to indicate direction for one-dimensional motiondirection for one-dimensional motion

Units are meters (m) in SI, Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in centimeters (cm) in cgs or feet (ft) in US CustomaryUS Customary

DisplacementsDisplacements

DistanceDistance

Distance may be, but is not necessarily, Distance may be, but is not necessarily, the magnitude of the displacementthe magnitude of the displacement

Blue line shows the distanceBlue line shows the distance Red line shows the displacementRed line shows the displacement

VelocityVelocity

It takes time for an object to It takes time for an object to undergo a displacementundergo a displacement

The The average velocityaverage velocity is rate at is rate at which the displacement occurswhich the displacement occurs

generally use a time interval, so let generally use a time interval, so let ttii = 0 = 0

t

xx

t

xv ifaverage

Velocity continuedVelocity continued

Direction will be the same as the Direction will be the same as the direction of the displacement (time direction of the displacement (time interval is always positive)interval is always positive) + or - is sufficient+ or - is sufficient

Units of velocity are m/s (SI), cm/s Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.)(cgs) or ft/s (US Cust.) Other units may be given in a Other units may be given in a

problem, but generally will need to be problem, but generally will need to be converted to theseconverted to these

SpeedSpeed

Speed is a scalar quantitySpeed is a scalar quantity same units as velocitysame units as velocity total distance / total timetotal distance / total time

May be, but is not necessarily, the May be, but is not necessarily, the magnitude of the velocitymagnitude of the velocity

Instantaneous VelocityInstantaneous Velocity

The limit of the average velocity as The limit of the average velocity as the time interval becomes the time interval becomes infinitesimally short, or as the time infinitesimally short, or as the time interval approaches zerointerval approaches zero

The instantaneous velocity The instantaneous velocity indicates what is happening at indicates what is happening at every point of timeevery point of time

Uniform VelocityUniform Velocity

Uniform velocity is constant Uniform velocity is constant velocityvelocity

The instantaneous velocities are The instantaneous velocities are always the same always the same All the instantaneous velocities will All the instantaneous velocities will

also equal the average velocityalso equal the average velocity

Graphical Interpretation of Graphical Interpretation of VelocityVelocity

Velocity can be determined from a Velocity can be determined from a position-time graphposition-time graph

Average velocity equals the slope of the Average velocity equals the slope of the line joining the initial and final positionsline joining the initial and final positions

Instantaneous velocity is the slope of Instantaneous velocity is the slope of the tangent to the curve at the time of the tangent to the curve at the time of interestinterest

The instantaneous speed is the The instantaneous speed is the magnitude of the instantaneous velocitymagnitude of the instantaneous velocity

Average VelocityAverage Velocity

Instantaneous VelocityInstantaneous Velocity

AccelerationAcceleration

Changing velocity (non-uniform) Changing velocity (non-uniform) means an acceleration is presentmeans an acceleration is present

Acceleration is the rate of change Acceleration is the rate of change of the velocityof the velocity

Units are m/s² (SI), cm/s² (cgs), Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust) and ft/s² (US Cust)

t

vv

t

va ifaverage

Average AccelerationAverage Acceleration

Vector quantityVector quantity When the sign of the velocity and When the sign of the velocity and

the acceleration are the same the acceleration are the same (either positive or negative), then (either positive or negative), then the speed is increasingthe speed is increasing

When the sign of the velocity and When the sign of the velocity and the acceleration are in the the acceleration are in the opposite directions, the speed is opposite directions, the speed is decreasingdecreasing

Instantaneous and Instantaneous and Uniform AccelerationUniform Acceleration

The limit of the average acceleration The limit of the average acceleration as the time interval goes to zeroas the time interval goes to zero

When the instantaneous When the instantaneous accelerations are always the same, accelerations are always the same, the acceleration will be uniformthe acceleration will be uniform The instantaneous accelerations will all The instantaneous accelerations will all

be equal to the average accelerationbe equal to the average acceleration

Graphical Interpretation of Graphical Interpretation of AccelerationAcceleration

Average acceleration is the slope Average acceleration is the slope of the line connecting the initial of the line connecting the initial and final velocities on a velocity-and final velocities on a velocity-time graphtime graph

Instantaneous acceleration is the Instantaneous acceleration is the slope of the tangent to the curve of slope of the tangent to the curve of the velocity-time graphthe velocity-time graph

Average AccelerationAverage Acceleration

Relationship Between Relationship Between Acceleration and VelocityAcceleration and Velocity

Uniform velocity (shown by red arrows Uniform velocity (shown by red arrows maintaining the same size)maintaining the same size)

Acceleration equals zeroAcceleration equals zero

Relationship Between Relationship Between Velocity and AccelerationVelocity and Acceleration

Velocity and acceleration are in the same directionVelocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the Acceleration is uniform (blue arrows maintain the

same length)same length) Velocity is increasing (red arrows are getting longer)Velocity is increasing (red arrows are getting longer)

Relationship Between Relationship Between Velocity and AccelerationVelocity and Acceleration

Acceleration and velocity are in opposite directionsAcceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the Acceleration is uniform (blue arrows maintain the

same length)same length) Velocity is decreasing (red arrows are getting Velocity is decreasing (red arrows are getting

shorter)shorter)

Kinematic EquationsKinematic Equations

Used in situations with uniform Used in situations with uniform accelerationacceleration

xa2vv

at2

1tvx

atvv

t2

vvtvx

2o

2f

2o

of

foaverage

Notes on the equationsNotes on the equations

t2

vvtvx fo

average

Gives displacement as a function Gives displacement as a function of velocity and timeof velocity and time

Notes on the equationsNotes on the equations

Shows velocity as a function of Shows velocity as a function of acceleration and timeacceleration and time

atvv of

Graphical Interpretation of Graphical Interpretation of the Equationthe Equation

Notes on the equationsNotes on the equations

Gives displacement as a function Gives displacement as a function of time, velocity and accelerationof time, velocity and acceleration

2o at

2

1tvx

Notes on the equationsNotes on the equations

Gives velocity as a function of Gives velocity as a function of acceleration and displacementacceleration and displacement

xa2vv 2o

2f

Problem-Solving HintsProblem-Solving Hints

Be sure all the units are consistentBe sure all the units are consistent Convert if necessaryConvert if necessary

Choose a coordinate systemChoose a coordinate system Sketch the situation, labeling initial and Sketch the situation, labeling initial and

final points, indicating a positive final points, indicating a positive directiondirection

Choose the appropriate kinematic Choose the appropriate kinematic equationequation

Check your resultsCheck your results

Free FallFree Fall

All objects moving under the influence All objects moving under the influence of only gravity are said to be in free fallof only gravity are said to be in free fall

All objects falling near the earth’s All objects falling near the earth’s surface fall with a constant accelerationsurface fall with a constant acceleration

Galileo originated our present ideas Galileo originated our present ideas about free fall from his inclined planesabout free fall from his inclined planes

The acceleration is called the The acceleration is called the acceleration due to gravity, and acceleration due to gravity, and indicated by gindicated by g

Acceleration due to Acceleration due to GravityGravity

Symbolized by gSymbolized by g g = 9.8 m/s²g = 9.8 m/s² g is always directed downwardg is always directed downward

toward the center of the earthtoward the center of the earth

Free Fall -- an object Free Fall -- an object droppeddropped

Initial velocity is Initial velocity is zerozero

Let up be positiveLet up be positive Use the kinematic Use the kinematic

equationsequations Generally use y Generally use y

instead of x since instead of x since verticalvertical

vo= 0

a = g

Free Fall -- an object Free Fall -- an object thrown downwardthrown downward

a = ga = g Initial velocity Initial velocity 0 0

With upward With upward being positive, being positive, initial velocity will initial velocity will be negativebe negative

Free Fall -- object thrown Free Fall -- object thrown upwardupward

Initial velocity is Initial velocity is upward, so positiveupward, so positive

The instantaneous The instantaneous velocity at the velocity at the maximum height is maximum height is zerozero

a = g everywhere a = g everywhere in the motionin the motion g is always g is always

downward, downward, negativenegative

v = 0

Thrown upward, cont.Thrown upward, cont.

The motion may be symmetricalThe motion may be symmetrical then tthen tupup = t = tdowndown

then vthen vff = -v = -voo

The motion may not be The motion may not be symmetricalsymmetrical Break the motion into various partsBreak the motion into various parts

generally up and downgenerally up and down

Non-symmetrical Non-symmetrical Free FallFree Fall

Need to divide the Need to divide the motion into motion into segmentssegments

Possibilities includePossibilities include Upward and Upward and

downward portionsdownward portions The symmetrical The symmetrical

portion back to the portion back to the release point and release point and then the non-then the non-symmetrical portionsymmetrical portion

Combination MotionsCombination Motions