VelocityVelocity

SpeedSpeed

SpeedSpeed measures the rate of change of position along measures the rate of change of position along a path.a path.

The direction doesn’t matter for speed, but path does.The direction doesn’t matter for speed, but path does.

9 m/s

12 m/s

3 m/s

The start of the arrows all occur at the same time.

The ends of the arrows all occur at the same time.

Longer arrows mean faster motion.

Not straight – not vectors.

VelocityVelocity

VelocityVelocity measures the rate of change of position, and measures the rate of change of position, and specifies the direction.specifies the direction.

The rate of change of position is similar for speed The rate of change of position is similar for speed and velocity.and velocity.

5 m/s northwest

8 m/s west

3 m/s northeast

Velocity has speed and direction

Velocity is a vector.

12

12

tt

rr

t

rv

Velocity MagnitudeVelocity Magnitude

Displacement divided by Displacement divided by time gives the average time gives the average velocity.velocity.

The displacement vector The displacement vector divided by time gives the divided by time gives the average velocity vector.average velocity vector.

A person walks from 2 km north A person walks from 2 km north of the gym to 3 km west of the of the gym to 3 km west of the gym in 1.5 h. What is the gym in 1.5 h. What is the magnitude of average velocity?magnitude of average velocity?

• The magnitude of The magnitude of displacement isdisplacement is

• The average velocity is the The average velocity is the magnitude of displacement magnitude of displacement divided by the time.divided by the time.

t

rvav

km 3.622 yx ddd

km/h 2.4/ tdvav

Graphing MotionGraphing Motion

Kinematics is the study of motion.Kinematics is the study of motion. Motion requires a change in position.Motion requires a change in position. We graph position as a function of time.We graph position as a function of time.

Which point is farthest?Which point is farthest?

Which point is fastest?Which point is fastest?

Which point is slowest?Which point is slowest?

In one dimension objects In one dimension objects only move on a straight line.only move on a straight line.

Objects can go forward and Objects can go forward and backward.backward.

Objects can speed up and Objects can speed up and slow down.slow down.

Motion in One DimensionMotion in One Dimension

Position (x) axis is verticalUnits on this

graph are relative

From P1 to P2 the position increases, and the velocity is positive.

From P3 to P4 the position decreases, and the velocity is negative.

Average VelocityAverage Velocity

The ratio The ratio vv = = xx / / tt gives gives the average velocity during the average velocity during the time interval the time interval tt..

In the graph,In the graph, xx = = xx22 – – xx11

tt = = tt22 – – tt11

This is the slope of the line.This is the slope of the line.

12

12

tt

xx

t

xv

Short TimesShort Times

If the time scale If the time scale changes the velocity changes the velocity may also change.may also change.

The average velocity The average velocity from Pfrom P11 to P to P22 (green) is (green) is

greater than the greater than the average velocity from Paverage velocity from P11

to Pto PAA (orange). (orange).

xA

tA

PA

As As tt22 gets closer to gets closer to tt11, ,

tt gets close to 0. gets close to 0.

The above expression The above expression cannot be calculated cannot be calculated at at tt = 0. = 0.

The limit is needed.The limit is needed.

LimitsLimits

A mathematical A mathematical limitlimit describes the result of taking describes the result of taking a value to an extreme.a value to an extreme.

This is used when the result This is used when the result at the extreme point would at the extreme point would give a mathematical give a mathematical expression that cannot be expression that cannot be calculated.calculated.

t

xv

t

xv

t

0

lim

Instananeous VelocityInstananeous Velocity

The limit of the velocity for The limit of the velocity for short times is the short times is the instantaneous velocityinstantaneous velocity..

The instantaneous velocity The instantaneous velocity appears as the slope of the appears as the slope of the graph at a point.graph at a point.

This black line is tangent to the curve at P1.

The slope of that line is the instantaneous velocity at P1.

Motion in Two DimensionsMotion in Two Dimensions

Time is a scalar, not a Time is a scalar, not a vector.vector.

y

x2r1r

t

rrv

12

y

x

2t

1t12 ttt

Vector average velocity is Vector average velocity is parallel to the displacement.parallel to the displacement.

t

yyv

t

xxv

y

x

12

12

TangentTangent

The average velocity The average velocity becomes the instantaneous becomes the instantaneous velocity for short time velocity for short time intervals.intervals.

The same is true for vectors.The same is true for vectors.

The instantaneous velocity The instantaneous velocity vector direction is tangent to vector direction is tangent to the curve.the curve.

y

x2r

1r v

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