Lecture IILecture II

General Physics (PHY 2170)

• Motion in one dimension Position and displacement Velocity

average instantaneous

Acceleration motion with constant acceleration

Lightning ReviewLightning Review

Last lecture:

1. Physics introduction: units, measurements, etc.

Review Problem:

How many beats would you detect if you take someone’s pulse for 10 sec instead of a minute? Hint: Normal heartbeat rate

is 60 beats/minute.

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 and relationship between that motion and other physics conceptsother physics concepts

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

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

motionmotion

Position and DisplacementPosition and Displacement► PositionPosition is defined in terms is defined in terms

of a of a frame of referenceframe of reference

Frame A: Frame A: xxii>0 >0 and and xxff>0 >0

Frame B: Frame B: x’x’ii<0 <0 butbut x’ x’ff>0 >0

► One dimensional, so One dimensional, so generally the generally the x- or y-axisx- or y-axis

A

By’

x’O’ xi’ xf’

Position and DisplacementPosition and Displacement► PositionPosition is defined in is defined in

terms of a terms of a frame of frame of referencereference One dimensional, so One dimensional, so

generally the generally the x- or y-axisx- or y-axis► DisplacementDisplacement measures measures

the the changechange in position in position Represented as Represented as xx (if (if

horizontal) or horizontal) or yy (if (if vertical)vertical)

Vector quantity (i.e. needs Vector quantity (i.e. needs directional information)directional information)

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

UnitsUnits

SISI Meters (m)Meters (m)

CGSCGS Centimeters (cm)Centimeters (cm)

US CustUS Cust Feet (ft)Feet (ft)

DisplacementDisplacement DisplacementDisplacement measures measures the the change in positionchange in position

represented as represented as xx or or yy

m

mm

xxx if

70

1080

1

m

mm

xxx if

60

8020

2

Distance or Displacement?Distance or Displacement?

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

Distance(blue line)

Displacement (yellow line)

Position-time graphsPosition-time graphs

Note: position-time graph is not necessarily a straight line, even though the motion is along x-direction

ConcepTest 1ConcepTest 1

An object (say, car) goes from one point in space to another. After it arrives to its destination, itsdisplacement is

1. either greater than or equal to 2. always greater than3. always equal to4. either smaller or equal to5. either smaller or larger

than the distance it traveled.

ConcepTest 1 (answer)ConcepTest 1 (answer)

An object (say, car) goes from one point in space to another. After it arrives to its destination, itsdisplacement is

1. either greater than or equal to 2. always greater than3. always equal to4. either smaller or equal to5. either smaller or larger

than the distance it traveled.

Note: displacement is a vector from the final to initial points, distance is total path traversed

Average VelocityAverage Velocity► It takes time for an object to undergo a It takes time for an object to undergo a

displacementdisplacement► The The average velocityaverage velocity is is raterate at which the at which the

displacement occursdisplacement occurs

► DirectionDirection will be will be the same asthe same as the direction of the direction of the the displacementdisplacement ( (tt is always positive) is always positive)

t

xx

t

xv if

average

More About Average VelocityMore About Average Velocity

►Units of velocity:Units of velocity:

►Note:Note: other units may be given in a other units may be given in a problem, problem, but generally will need to be but generally will need to be converted to theseconverted to these

UnitsUnits

SISI Meters per second (m/s)Meters per second (m/s)

CGSCGS Centimeters per second (cm/s)Centimeters per second (cm/s)

US CustomaryUS Customary Feet per second (ft/s)Feet per second (ft/s)

Example:Example:

sms

m

t

xv average

710

7011

Suppose that in both cases truck covers the distance in 10 seconds:

sms

m

t

xv average

610

6022

SpeedSpeed

►Speed is a Speed is a scalarscalar quantity (no quantity (no information about sign/direction is information about sign/direction is need)need) same units as velocitysame units as velocity Average speed = total distance / total Average speed = total distance / total

timetime

►Speed is the magnitude of the velocitySpeed is the magnitude of the velocity

Graphical Interpretation of Average Graphical Interpretation of Average VelocityVelocity

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

► Average velocityAverage velocity equals the equals the slopeslope of the of the line joining the initial and final positionsline joining the initial and final positions

sms

m

t

xvaverage

130.3

40

Instantaneous VelocityInstantaneous Velocity

► Instantaneous velocityInstantaneous velocity is defined as the is defined as the limit of the average velocitylimit of the average velocity as the time as the time interval becomes infinitesimally short, interval becomes infinitesimally short, or as the time interval approaches zeroor as the time interval approaches zero

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

0 0lim lim f i

instt t

x xx dxv

t t dt

Uniform VelocityUniform Velocity

►UniformUniform velocity is velocity is constantconstant velocity velocity►The instantaneous velocities are The instantaneous velocities are

always the same always the same All the instantaneous velocities will also All the instantaneous velocities will also

equal the average velocityequal the average velocity

Graphical Interpretation of Graphical Interpretation of Instantaneous VelocityInstantaneous Velocity

► Instantaneous velocityInstantaneous velocity is the is the slopeslope of the of the tangenttangent to the curve at the time of to the curve at the time of interestinterest

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

Average vs Instantaneous VelocityAverage vs Instantaneous Velocity

Average velocity Instantaneous velocity

ConcepTest 2ConcepTest 2

The graph shows position as a function of timefor two trains running on parallel tracks. Which of the following is true:

1. at time tB both trains have the same velocity 2. both trains speed up all the time3. both trains have the same velocity at some time before tB

4. train A is longer than train B5. all of the above statements are true

A

B

time

position

tB

ConcepTest 2 (answer)ConcepTest 2 (answer)

The graph shows position as a function of timefor two trains running on parallel tracks. Which of the following is true:

1. at time tB both trains have the same velocity 2. both trains speed up all the time3. both trains have the same velocity at some time before tB

4. train A is longer than train B5. all of the above statements are true

Note: the slope of curve B is parallel to line A at some point t< tB

A

B

time

position

tB

Average AccelerationAverage Acceleration► Changing velocity (non-uniform) means Changing velocity (non-uniform) means

an acceleration is presentan acceleration is present► Average accelerationAverage acceleration is the is the rate of rate of

change of the velocitychange of the velocity

► Average acceleration is a Average acceleration is a vectorvector quantity quantity (i.e. described by both magnitude and (i.e. described by both magnitude and direction)direction)

t

vv

t

va if

average

Average AccelerationAverage Acceleration

► When the When the signsign of the of the velocityvelocity and the and the accelerationacceleration are the are the samesame (either (either positive or negative), then positive or negative), then the speed is the speed is increasingincreasing

► When the When the signsign of the of the velocityvelocity and the and the accelerationacceleration are are oppositeopposite, , the speed is the speed is decreasingdecreasing

UnitsUnits

SISI Meters per second squared (m/sMeters per second squared (m/s22))

CGSCGS Centimeters per second squared (cm/sCentimeters per second squared (cm/s22))

US CustomaryUS Customary Feet per second squared (ft/sFeet per second squared (ft/s22))

Instantaneous and Uniform Instantaneous and Uniform AccelerationAcceleration

► Instantaneous accelerationInstantaneous acceleration is the is the limitlimit of the average acceleration as the time of the average acceleration as the time interval goes to zerointerval goes to zero

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

equal to the average accelerationequal to the average acceleration

0 0lim lim f i

instt t

v vv dva

t t dt

Graphical Interpretation of Graphical Interpretation of AccelerationAcceleration

► Average accelerationAverage acceleration is is the the slopeslope of the line of the line connecting the connecting the initial and initial and final velocitiesfinal velocities on a on a velocity-time graphvelocity-time graph

► Instantaneous Instantaneous accelerationacceleration is the is the slopeslope of of the the tangenttangent to the curve of to the curve of the velocity-time graphthe velocity-time graph

One-dimensional Motion With One-dimensional Motion With Constant AccelerationConstant Acceleration

► If If acceleration is uniform acceleration is uniform (i.e. ):(i.e. ):

thus:thus:

atvv of

Shows Shows velocityvelocity as a function of as a function of accelerationacceleration and and timetime

t

vv

tt

vva of

f

of

0

aa

One-dimensional Motion With One-dimensional Motion With Constant AccelerationConstant Acceleration

► Used in situations with Used in situations with uniform accelerationuniform acceleration

tvv

tvx foaverage

2

21

2ox v t at Velocity changesuniformly!!!

2 2 2f ov v a x

atvv of

Notes on the equationsNotes on the equations

2o f

average

v vx v t t

► Gives displacement as a function of velocity and Gives displacement as a function of velocity and

timetime

► Gives displacement as a function of time, Gives displacement as a function of time, velocity and accelerationvelocity and acceleration

► Gives velocity as a function of acceleration and Gives velocity as a function of acceleration and displacementdisplacement

21

2ox v t at

2 2 2f ov v a x

Another look at ConstantAnother look at Constant AccelerationAcceleration

► Since let’s rewrite it Since let’s rewrite it as as

dxv

dt

Let’s integrate both sides:Let’s integrate both sides:

dx vdt

0 0

x t

x

dx v dt

Then, the result is:Then, the result is:

20 0

0

1

2

t

ox x v at dt v t at

ov v at

Just as before!Just as before!

Summary of kinematic Summary of kinematic equationsequations

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

►This acceleration is called the This acceleration is called the acceleration due to gravityacceleration due to gravity, and , and indicated by gindicated by g

Acceleration due to GravityAcceleration due to Gravity

► Symbolized by gSymbolized by g► g = 9.8 m/s² (can use g = 10 m/s² for g = 9.8 m/s² (can use g = 10 m/s² for

estimates)estimates)► 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

► InitialInitial velocity is velocity is zerozero► Frame: Frame: let up be positivelet up be positive► Use the kinematic Use the kinematic

equationsequations Generally use Generally use yy instead instead

of of xx since vertical since vertical

vo= 0

a = g

2

2

8.9

2

1

sma

aty

y

x

Free Fall -- an Object Thrown Free Fall -- an Object Thrown DownwardDownward

► a = ga = g With upward being positiveWith upward being positive, ,

acceleration will be acceleration will be negative,negative, g = -9.8 m/s² g = -9.8 m/s²

► Initial velocity Initial velocity 0 0 With upward being positiveWith upward being positive, ,

initial velocity will be initial velocity will be negativenegative

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

► Initial velocity is Initial velocity is upwardupward, so , so positivepositive

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

► a = g everywhere in a = g everywhere in the motionthe motion g is always g is always

downward, negativedownward, negative

v = 0

Thrown upwardThrown upward

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

then vthen vff = -v = -voo

►The motion may not be symmetricalThe motion may not be symmetrical Break the motion into various partsBreak the motion into various parts

►generally up and downgenerally up and down

ConcepTest 3ConcepTest 3

A person standing at the edge of a cliff throws oneball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown

1. upward 2. downward3. neither – they both hit at the same speed

ConcepTest 3 (answer)ConcepTest 3 (answer)

A person standing at the edge of a cliff throws oneball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit ground below the cliff with greater speed is the one initially thrown

1. upward 2. downward3. neither – they both hit at the same speed

Note: upon the descent, the velocity of an object thrown straightup with an initial velocity v is exactly –v when it passes the pointat which it was first released.

Fun QuickLab: Reaction timeFun QuickLab: Reaction time

g

dt

smgtgd

2

8.9,2

1 22