The positionposition of any object must be givenwith respect to some reference pointreference point.

An object’s positionposition is its directeddirecteddistancedistance from a reference pointreference point.

MovementMovementis said to have occurred when the positionposition of an object with respect to a given reference reference

pointpoint has changedchanged.

Displacement, then, is a Displacement, then, is a vectorvector quantity. quantity.

DistanceDistance, or length, is a , or length, is a scalarscalar..

Click Click here and here to explore differences between distancedistance and and

displacementdisplacement..

This change in position of an objectThis change in position of an objectis often called its is often called its displacementdisplacement..

It is the It is the straight line distance and straight line distance and directiondirection from the starting point to from the starting point to the ending point, regardless of the the ending point, regardless of the

path taken.path taken.

In one dimensional motion, the displacement directiondisplacement direction is often given

as positivepositive, +, or negativenegative, -.

A displacementdisplacement of ++3.5 m implies movement of 3.5 m in the positivepositive direction.

A displacementdisplacement of --3.5 m implies movementof 3.5 m in the negativenegative direction.

PositivePositive and negativenegative directions are

chosen arbitrarilyarbitrarily, but usually agree with

standard mathematical conventions.

The average velocityaverage velocity of an object is defined tobe the ratioratio of its change in positionchange in position to the

time taken to change the positiontime taken to change the position.

vvavavvvavav = average velocity;

in units of m/s, mph, ft/s, km/hrm/s, mph, ft/s, km/hr, etc...

= dd

dd = change in position, or displacement; in units of m, in, ft, km, mim, in, ft, km, mi, etc...

tt

tt = change in time; in units of s, min, hrs, min, hr, etc...

A positive signpositive sign indicates movementin the positive directionpositive direction.

A negative signnegative sign indicates movement in the negative directionnegative direction.

The “The “signsign” of the velocity indicates” of the velocity indicatesthe the direction of movementdirection of movement..

SpeedSpeed is the magnitude of velocity.

It is a scalarscalar and has no direction given with it.

Average speedAverage speed is the total distancetotal distance traveled divided bydivided by the total timetotal time taken.

Average velocityAverage velocity is the total displacementtotal displacement divided bydivided by the total timetotal time taken.

Average speed and average velocity areAverage speed and average velocity aregenerally not equivalent because total distancegenerally not equivalent because total distance

and total displacement are generally not the same.and total displacement are generally not the same.

When would they have the same magnitudes?When would they have the same magnitudes?

Instantaneous speed is the absolute valueabsolute value of the instantaneous velocity.It is always a positivepositive value.

If an object increasesincreases its speed while traveling in the negative directionnegative direction,

its velocity actually decreasesdecreases.

If an object If an object decreasesdecreases its speed while its speed while traveling in the traveling in the negative directionnegative direction, ,

its velocity actually its velocity actually increasesincreases..

The average accelerationaverage acceleration of an object is defined to be the ratioratio of its change in velocitychange in velocity to the

time taken to change the velocitytime taken to change the velocity.

aaavavaaavav = average acceleration;

in units of m/s/s, mph/s, ft/s/s, km/hr/sm/s/s, mph/s, ft/s/s, km/hr/s, etc...

= vv

vv = change in velocity; in units of m/s, in/s, ft/s, km/hr, mphm/s, in/s, ft/s, km/hr, mph, etc...

tt

tt = change in time; in units of s, min, hrs, min, hr, etc...

A positive signpositive sign indicates that the velocity is increasingincreasing. It will also be an increaseincrease inin

speedspeed if the object is traveling in the positive positive directiondirection. It is a decrease in speed otherwise.

A negative signnegative sign indicates that the velocityis decreasingdecreasing. It will also be a decrease indecrease in

speedspeed if the object is traveling in the positive positive directiondirection. It is an increase in speed otherwise.

The “The “signsign” of the acceleration” of the accelerationindicates whether the velocity isindicates whether the velocity is

increasingincreasing or or decreasingdecreasing..

It is important to note that information about an object’s It is important to note that information about an object’s acceleration tells us how the object’s velocity is changing.acceleration tells us how the object’s velocity is changing.

In order to know what this change in velocity In order to know what this change in velocity is doing to the object’s speed, we mustis doing to the object’s speed, we must

know the direction the object is traveling.know the direction the object is traveling.

As a rule, if the object’s velocity and accelerationAs a rule, if the object’s velocity and accelerationare in the same direction (have the same sign),are in the same direction (have the same sign),

we can say that the object’s speed is increasing.we can say that the object’s speed is increasing.

If the velocity and acceleration are in oppositeIf the velocity and acceleration are in oppositedirections (have opposite signs), we knowdirections (have opposite signs), we know

that the object’s speed is decreasing.that the object’s speed is decreasing.

What can we say about changes in speed and/or velocity if What can we say about changes in speed and/or velocity if the acceleration is either increasing or decreasing?the acceleration is either increasing or decreasing?

ConstantConstant

vvff = v = vii + at + atd = vd = vavavtt

vvavav = (v = (vff + v + vii)/2)/2

d = vd = viit + 0.5att + 0.5at22

vvff22 = v = vii

22 + 2ad + 2ad

t = timed = displacement

vi = initial velocityvav = average

velocitya = acceleration

vf = final velocity

You can learn more about distance,You can learn more about distance,displacement, speed, velocity, displacement, speed, velocity,

relative velocity, and accelerationrelative velocity, and accelerationby exploring these links:by exploring these links:

link1, link2, link3

Position – Time Graphs SummarizedPosition – Time Graphs Summarized

• the the y-coordinatey-coordinate at any time gives the at any time gives the positionposition of the object of the object

• the the slopeslope of a position-time graph at any instant is the instantaneous of a position-time graph at any instant is the instantaneous velocityvelocity of the object of the object

• horizontalhorizontal graph segments indicate that the object is “ graph segments indicate that the object is “at restat rest””

• graph segments moving graph segments moving upwardupward imply movement in the imply movement in the positive positive directiondirection

• graph segments moving graph segments moving downwarddownward imply movement in the imply movement in the negative negative direction direction

• straightstraight line graph segments indicate line graph segments indicate constant speedconstant speed

• curvingcurving graph segments indicate graph segments indicate changing speedchanging speed

• graph segments becoming graph segments becoming steepersteeper indicate an indicate an increase in speedincrease in speed

• graph segments becoming graph segments becoming less steepless steep indicate a indicate a decrease in speeddecrease in speed

• a a change of directionchange of direction is indicated whenever the graph is is indicated whenever the graph is concaveconcave upward or downwardupward or downward

Velocity – Time Graphs SummarizedVelocity – Time Graphs Summarized

• the the y-coordinatey-coordinate at any time gives the at any time gives the velocityvelocity of the object of the object• the the slopeslope of a velocity-time graph is the of a velocity-time graph is the accelerationacceleration of the object of the object• horizontalhorizontal graph segments indicate that the object has graph segments indicate that the object has constant constant velocityvelocity• graph segments graph segments above the x-axisabove the x-axis imply movement in the imply movement in the positive positive directiondirection• graph segments graph segments below the x-axisbelow the x-axis imply movement in the imply movement in the negative negative direction direction • horizontal segments horizontal segments on the x-axison the x-axis indicate no movement indicate no movement• straightstraight line graph segments indicate line graph segments indicate constant accelerationconstant acceleration• graph segments moving upward indicate an graph segments moving upward indicate an increase in velocityincrease in velocity• graph segments moving downward indicate a graph segments moving downward indicate a decrease in velocitydecrease in velocity• a a change of directionchange of direction is indicated whenever the graph crosses the x- is indicated whenever the graph crosses the x- axisaxis• an an increase in speedincrease in speed is indicated by graph segments moving is indicated by graph segments moving away away from the x-axisfrom the x-axis

Acceleration – Time Graphs SummarizedAcceleration – Time Graphs Summarized

• the the y-coordinatey-coordinate at any time gives the at any time gives the accelerationacceleration of the object of the object• horizontalhorizontal graph segments indicate that the object has graph segments indicate that the object has constant constant accelerationacceleration• a horizontal graph segment a horizontal graph segment on the x-axison the x-axis indicates that the object indicates that the object has has constant velocityconstant velocity (no acceleration) (no acceleration)• graph segments graph segments above the x-axisabove the x-axis imply imply increasing velocities increasing velocities • graph segments graph segments below the x-axisbelow the x-axis imply imply decreasing velocities decreasing velocities • no changes in direction may be inferred from these graphsno changes in direction may be inferred from these graphs

At the introductory physics level, we typically only deal At the introductory physics level, we typically only deal with constant acceleration situations, so acceleration with constant acceleration situations, so acceleration graphs generally consist of horizontal segments only.graphs generally consist of horizontal segments only.