physics 114a - mechanics lecture 6 (walker: 3.4-5) the r, v, and a vectors january 12, 2012 john g....

Physics 114A - MechanicsLecture 6 (Walker: 3.4-5)

The r, v, and a VectorsJanuary 12, 2012

Physics 114A - MechanicsLecture 6 (Walker: 3.4-5)

The r, v, and a VectorsJanuary 12, 2012

John G. CramerProfessor Emeritus, Department of Physics

B451 [email protected]

January 12, 2012 Physics 114A - Lecture 6 2/24

AnnouncementsAnnouncements Because of recent problems with WebAssign, I have moved the

due date for Assignment #1 to 11:59 PM on Tuesday, January 17. Now all 215 registered students have WebAssign Accounts. Good! Assignment #2 is still due at 11:59 PM on Thursday, January 19.

Register your clicker using the “Clicker” link on the Physics 114A Syllabus page. So far 188/215 students have registered their clickers. You may talk to your neighbors during a clicker session. If your clicker ID is not appearing, reset it to Channel 01 and/or if necessary, get closer to the pickup antenna.

We will have Exam 1 on Friday, January 20. It will be about 75% multiple choice and 25% free response. The questions will be based on Walker “2-dot” end-of-chapter questions. There will be assigned seating. If you wish a left-handed seat, a right-handed aisle seat, or a front row seat, send your request to me by E-mail.


Physics 114A - Introduction to Mechanics - Winter-2012

Lecture: Professor John G. Cramer

Textbook: Physics, (Edition: UW Vol. 1 or Complete 4th), James S. Walker

Week Date L# Lecture Topic Pages Slides Reading HW Due Lab

1

2-Jan-12 H1 New Year Holiday

No Lab 1st week3-Jan-12 1 Introduction to Physics 12 21 Chapter 1

5-Jan-12 2 Position & Velocity 8 22 2-1 to 2-3 No HW

6-Jan-12 3 Velocity & Acceleration 10 25 2-4 to 2-5

2

9-Jan-12 4 Equations of Motion 9 20 2-6 to 2-7

1-D Kinematics10-Jan-12 5 Vectors 8 24 3-1 to 3-3

12-Jan-12 6 r, v & a Vectors 5 24 3-4 to 3-5 HW1

13-Jan-12 7 Relative Motion 3 18 3-6

3

16-Jan-12 H2 MLK Birthhday Holiday

Free Fall & Projectiles17-Jan-12 8 2D Motion Basics 5 19 4-1 to 4-2

19-Jan-12 9 2D Examples 13 22 4-3 to 4-5 HW2

20-Jan-12 E1 EXAM 1 - Chapters 1-4

Lecture Schedule (Part 1)

Lecture Schedule (Part 1)

We are here.


Unit VectorsUnit Vectors Unit vectors are dimensionless vectors of unit length. For example, the vector has magnitude (with units) Ax and points along the +x axis.

ˆxA A x

A


Scalar-Vector MultiplicationScalar-Vector MultiplicationMultiplying vectors

by scalars: the scalar multiplier changes the vector length, and the scalar sign can reverse the direction.

This property allows vectors to be factored into a unit vector and a multiplier. The unit vector specifies only the direction, while the multiplier provides the magnitude and units of the vector.


Unit VectorsUnit Vectorsˆ (1,0,0) unit vector in +x-direction = "x-hat"

ˆ (0,1,0) unit vector in +y-direction = "y-hat"

ˆ (0,0,1) unit vector in +z-direction = "z-hat"

x

y

z

ˆ ˆ ˆ ( , , )x y k x y z x y zA A A A A x A y A z A A A

ˆ ˆ ˆ4 2 5 (4, 2,5)B x y z Example:

An alternative notation is:

ˆˆ ˆˆ ˆ ˆ, , and . x i y j z k


Position, Displacement, Velocity, & Acceleration

Vectors


Vectors The position vector r points from the origin to the location in question. The displacement vector Δr points from the original position to the final position.


The Displacement VectorThe Displacement Vector

ˆ ˆr xx yy

2 1r r r

2 1

2 2 1 1ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ

r r r

x x y y x x y y

xx y y


The Average Velocity VectorThe Average Velocity Vector

Average velocity vector:

(3-3)

So vav is in the same

direction as Δr.


The Average & Instantaneous Velocity

Vectors

The Average & Instantaneous Velocity

VectorsInstantaneous velocity vector is tangent to the path:


Velocity VectorsVelocity Vectors

2 1

2 2 1 1ˆ ˆ ˆ ˆ( ) ( )

ˆ ˆ

r r r

x x y y x x y y

xx y y

av

rv

t

0limt

rv

t

0 0 0 0

ˆ ˆˆ ˆlim lim lim lim

t t t t

r xx y y x yv x y

t t t t

ˆ ˆx yv v x v y 2 2 ; arctan y

x yx

vv v v

v


Example: A DragonflyExample: A Dragonfly

A 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

2 ˆ ˆ( 3.00 m) (5.50 m)r x y

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)

av

r rrv

t tx y

x y


110 m 130 m0.167 m/s

120 sxav

xv

t

Example: Velocity of a Sailboat

Example: Velocity of a Sailboat

ˆ ˆav xav yavv v x v y

218 m 205 m0.108 m/s

120 syav

yv

t

ˆ ˆ( 0.167 m/s) (0.108 m/s)avv x y

2 2( 0.167 m/s) (0.108 m/s) 0.199 m/savv

o0.108 m/sarctan 147

0.167 m/s

A 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 ;

avv

avv


Velocity & Acceleration Vectors

Velocity & Acceleration Vectors

The average acceleration vector a is in the direction of the change in velocity v:


Acceleration on a CurveAcceleration on a Curve

Typically, for a vehicle moving on a curve at a uniform speed, the acceleration is perpendicular to the velocity and the magnitude of the velocity stays the same, while the velocity’s direction changes.

For example, consider a car that has an initial velocity of 12 m/s east, 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 y

t


Example: Car Acceleratingon a Curve

Example: Car Acceleratingon a Curve

A car is traveling northwest at 9.00 m/s. Eight seconds later it has rounded a corner and is now headed north at 15.0 m/s/(a) What is the magnitude and direction of the acceleration during those 8.0 s?

N (y)

E(x)

ˆ ˆ ˆ ˆ(9.0 m/s)cos135 (9.0 m/s)sin135 ( 6.36 m/s) (6.36 m/s)iv x y x y

ˆ(15.0 m/s)fv y

ˆ ˆ(6.36 m/s) (8.64 m/s)f iv v v x y

2 2ˆ ˆ(0.795 m/s ) (1.08 m/s )av

va x y

t

2 2 2 2 2(0.795 m/s ) (1.08 m/s ) 1.34 m/sava

1 2 2tan (1.08 m/s ) / (0.795 m/s ) 53.6a


Acceleration VectorsAcceleration Vectors

;av

va

t

0limt

va

t

0ˆ ˆ ˆ ˆˆ ˆlimx y z

t

x y zv v x v y v z x y z

t t t

0ˆ ˆ ˆ ˆˆ ˆlim yx z

x y zt

vv va x y z a x a y a z

t t t

0 0 0lim ; lim ; limyx z

x y zt t t

vv va a a

t t t



Vectors


Vectors The velocity vector v always points in the direction of motion.

The acceleration vector a can point anywhere.


Clicker Question 1Clicker Question 1

A particle moves to the right in the x-y plane along the blue path. Identify each vector in the figure as a position (P) , velocity (V), or acceleration (A) vector.

(1) 1=V, 2=A, 3=P, 4=V; (2) 1=A, 2=P, 3=V, 4=P;

(3) 1=P, 2=V, 3=V, 4=A; (4) 1=P, 2=A, 3=A, 4=V;


Example: A Thrown Baseball

Example: A Thrown Baseball

The position of a thrown baseball is given by:

(a) Find the velocity as a function of time.(b) Find the acceleration as a function of time.

2 2ˆ ˆ[1.5 m (12 m/s) ] [(16 m/s) (4.9 m/s ) ]r t x t t y

1.5 m (12 m/s) ;x t 2 2(16 m/s) (4.9 m/s )y t t

0lim 12 m/s;xt

xv

t

2

0lim (16 m/s) 2(4.9 m/s )y t

yv t

t

0lim 0;x

x t

va

t

2

0lim 9.8 m/sy

y t

va

t

2ˆ ˆ(12 m/s) [(16 m/s) (9.8 m/s ) ] ;v x t y 2 ˆ( 9.8 m/s )a y


Example: Rounding a Curve

Example: Rounding a Curve

A car is traveling east at 60 km/h. It rounds a curve, and 5.0 s later it is traveling north at 60 km/h.(a) Find the average acceleration of the car.

;av

va

t

f iv v v

ˆ ˆ(60 km/h) (60 km/h)

5.0 sf i

av

v v j ia

t

1 h 1000 m60 km/h 16.7 m/s

3600 s 1 km

2 2ˆ ˆ(16.7 m/s) (16.7 m/s)ˆ ˆ( 3.4 m/s ) (3.4 m/s )

5.0 sav

y xa x y


Direction of theAcceleration Vector

Direction of theAcceleration Vector

Bungee Jumper’sDecent and Ascent

Bungee Jumper


You are asked to substitute for an illperformer in a circus that is sponsoredby your school. The job, should youchoose to accept it, is to be shotout of a cannon. Never afraidto accept a challenge, youaccept. The barrel ofthe cannon is inclinedat an angle of 600

above the horizontal. Your physics teacher offers you extra credit if you successfully use a motion diagram to estimate your acceleration during the ascending portion of your flight. 1. Make a motion diagram with velocities. 2. Draw a vector addition diagram usingthe relation: 3. Draw the acceleration from:

Example: Human Cannonball

Example: Human Cannonball

i fv v v

/a v t

Before the next lecture, read WalkerChapter 3.6

Homework Assignment #1 should be submitted using the WebAssign system by11:59 PM on Thursday, January 12 (Tonight!). Homework Assignment #2 should be submitted using the WebAssign system by 11:59 PM on Thursday, January 19.

End of Lecture 6End of Lecture 6

physics 114a - mechanics lecture 6 (walker: 3.4-5) the r, v, and a vectors january 12, 2012 john g....

Documents

lecture schedule

vector length

velocity vectorsphysics

velocity acceleration10252

position vector r points

scalar multiplier

unit vectorsphysics

unit vectorsexample