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Physics 111.01Lecture 6 (Walker: 3.4-3.6)

Unit VectorsPosition, Displacement, Velocity, & Acceleration

Vector EquationsRelative Motion

February 9, 2009

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Unit VectorsA unit vector is a symbol for a certain defined

direction, such as “North”, or “along the x axis”.A unit vector has magnitude equal to 1 and no units, but

has a defined direction.

Example: the unit vector in the x direction is usually written:

Now, I can write a vector as a sum of scalar components multiplied by the corresponding unit vector:

where Ax and Ay are scalar components (with units).

x

yxA ˆˆ yx AA +=

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Unit Vectors

Example of vector written in terms of scalar components and unit vectors:

xy

ymxmr ˆ)3(ˆ)2( +=r

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Motion in Two DimensionsMotion in Two Dimensions

►►Using + or Using + or –– signs for direction is usually signs for direction is usually notnot sufficient to fully describe motion in sufficient to fully describe motion in more than one dimensionmore than one dimension

►►VectorsVectors are used to fully describe motionare used to fully describe motion►►Interested in displacement, velocity, and Interested in displacement, velocity, and

acceleration vectorsacceleration vectors

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DisplacementDisplacement►► The position of an The position of an

object is described by object is described by its its position vectorposition vector, , rr

►► The The displacementdisplacement of of the object is defined as the object is defined as the the change in its change in its positionposition

∆∆rr = = rrff -- rrii

yyxxr ˆˆ +=r

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The Displacement Vector

ˆ ˆr xx yy= +r

2 1r r r∆ = −r r r

2 1

2 2 1 1ˆ ˆ ˆ ˆ( ) ( )ˆ ˆ

r r rx x y y x x y y

xx y y

∆ = −= + − +

= ∆ + ∆

r r r

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VelocityVelocity►► The The average velocityaverage velocity is the ratio of the displacement to is the ratio of the displacement to

the time interval for the displacementthe time interval for the displacement

The direction of The direction of vvavav is the direction of is the direction of ∆∆rr►► The The instantaneous velocityinstantaneous velocity is the limit of the average is the limit of the average

velocity as velocity as ∆∆t approaches zerot approaches zeroThe The directiondirection of the instantaneous velocity is along a of the instantaneous velocity is along a lineline tangent to the pathtangent to the path of the particle and in the of the particle and in the direction of motiondirection of motion

0limt

rvt∆ →

∆=

∆

rr

trvav ∆

∆=

rr

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The Average Velocity VectorAverage velocity vector:

(3-3)

So vav is in the same

direction as ∆r.

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Example: A DragonflyA dragonfly is observed initially at position:

Three seconds later, it is observed at position:

What was the dragonfly’s average velocity during this time?

1 ˆ ˆ(2.00 m) (3.50 m)r x y= +r

2 ˆ ˆ( 3.00 m) (5.50 m)r x y= − +r

[ ] [ ]

2 1

ˆ ˆ( 3.00 m) (2.00 m) (5.50 m) (3.50 m)(3.00 s)

ˆ ˆ( 1.67 m/s) (0.667 m/s)

avr rrv

t tx y

x y

−∆= =∆ ∆− − + −

=

= − +

r rrr

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110 m 130 m 0.167 m/s120 sxav

xvt

∆ −= = = −∆

Example: Velocity of a Sailboat

ˆ ˆav xav y avv v x v y= +r

218 m 205 m 0.108 m/s120 sy av

yvt

∆ −= = =∆

ˆ ˆ( 0.167 m/s) (0.108 m/s)avv x y= − +r

2 2( 0.167 m/s) (0.108 m/s) 0.199 m/savv = − + =

o0.108 m/sarctan 1470.167 m/s

θ = =−

Sailboat has coordinates (130 m, 205 m) at t1=0.0 s. Two minutes later its position is (110 m, 218 m). (a) Find ; (b) Find avvr avvvav vav

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AccelerationAcceleration►► The The average accelerationaverage acceleration is defined as the rate is defined as the rate

at which the velocity changesat which the velocity changes

►► The The instantaneous accelerationinstantaneous acceleration is the limit of is the limit of the average acceleration as the average acceleration as ∆∆t approaches zerot approaches zero

0limt

vat∆ →

∆=

∆

rr

tvaav ∆

∆=

rr

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Ways an Object Might AccelerateWays an Object Might Accelerate

►► The The magnitude of the velocitymagnitude of the velocity (the speed) can change(the speed) can change

►► The The direction of the velocitydirection of the velocity can changecan changeThen there is acceleration, even though the magnitude is Then there is acceleration, even though the magnitude is constant constant ---- example: motion in a circle at constant speedexample: motion in a circle at constant speed

►► Both the magnitude and the directionBoth the magnitude and the direction can changecan change

0limt

vat∆ →

∆=

∆

rr

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Position, Displacement, Velocity, & Acceleration Vectors

Velocity vector v always points in the direction of motion.The acceleration vector a can point anywhere.

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Acceleration on a CurveAcceleration on a Curve

Typically, for a vehicle moving on a curve, the magnitude of the velocity stays the same but the direction changes.

For example, consider a car that has an initial velocity of 12 m/seast, and 10 seconds later its velocity is 12 m/s south.

2 2ˆ ˆ( 12 m/s) (12 m/s) ˆ ˆ( 1.2 m/s ) ( 1.2 m/s )(10.0 s)av

v y xa x yt

∆ − −= = = − + −∆

rr

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Vector Motion withConstant Acceleration

Average velocity:1

02 ( )avv v v= +r r r

10 0 02( ) ( )avr t r v t r v v t= + = + +

r r r r r r

1 20 0 2( )r t r v t at= + +

r r r r

Position as a function of time:

Velocity as a function of time:

0( )v t v at= +r r r

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Relative MotionSuppose you swim downstream at 3 mi/hr and the current is moving at 2 mi/hr. What is your speed relative to the bank of the river?

Velocities must be added as vectors.

+ =vsw = 3 mi/hr

velocity of swimmer relative to water

velocity of water relative to bank

vwb = 2 mi/hr vsb = 5 mi/hr

velocity of swimmer relative to bank

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Relative Motion

In general

v13 = v12 + v23

velocity of 1 relative to 3

velocity of 1 relative to 2

velocity of 2 relative to 3

Note that v12 = –v21

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Example

A man is walking toward the back of a train at 2 m/s relative to the train, which is heading North. His velocity relative to the ground is 1.1 m/s North. What is the velocity of the train relative to the ground?

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Relative MotionThe speed of the passenger with respect to

the ground depends on the relative directions of the passenger’s and train’s speeds:

vpg = vpt + vtg = 16.2 m/s x vpg = vpt + vtg = 13.8 m/s x

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Relative Motion

This also works in two dimensions:

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Example: Flying a Plane

pG pA AGv v v= +r r r

A pilot wishes to fly a plane due north relative to the ground. The airspeed of the plane is 200 km/h, and the wind is blowing from west to east at 90 km/h.(a) In which direction should the plane head?(b) What will be the ground speed of the plane?

(90 km/h)arcsin arcsin 26.7 west of north(200 km/h)

AG

pA

vv

θ = = = °

2 2 2 2(200 km/h) (90 km/h) 179 km/hpG pA AGv v v= − = − =

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Example: Crossing a RiverYou are in a boat with a speed relative to the water of vbw = 6.1 m/s. Boat points at angle θ = 25°upstream on a river flowing at vwg = 1.4 m/s.(a) What is your speed vbg and angle θbg relative to the ground?

bg bw wgv v v= +r r r

ˆ( 1 .4 m /s)w gv y= −r

ˆ ˆ(6.1 m/s)cos 25 (6.1 m/s)sin 25ˆ ˆ(5.5 m/s) (2.6 m/s)

bwv x yx y

= ° + °= +

r

ˆ ˆ(5.5 m/s) (2.6 m/s 1.4 m/s)ˆ ˆ(5.5 m/s) (1.2 m/s)

bgv x y

x y

= + −

= +

r

2 2(5.5 m/s) (1.2 m/s) 5.6 m/sbgv = + =

[ ]1tan (1.2 m/s) / (5.5 m/s) 12bgθ −= = °

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Before the next lecture, read Walker 4.1-3

Homework Assignments #3b on WebAssignis due by 11:00 p.m. on Wed. Feb. 11.

Key Points of Lecture 6

Unit Vectors x , y

Vector equations for Displacement, Velocity, and Acceleration

Relative Motion v13 - v12 + v23