Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 1 PHYS 1443 – Section 501...

Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt

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PHYS 1443 – Section 501Lecture #4

Monday, Feb. 2, 2004Dr. Andrew Brandt

Motion in Two DimensionsVector Properties and OperationsMotion under constant acceleration


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Announcements• Homework: 38/44 of you have signed up (11

have submitted some answers)• Keep enough significant figures, get sign right • HW1 dues Weds at midnight (questions before

class?)


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Kinetic Equations of Motion in a Straight Line Under Constant Acceleration

tavtv xxixf

2

2

1tatvxx xxiif

tvvtvxx xixfi xf 2

1

2

1

ifxf xxavv xxi 222

Velocity as a function of time

Displacement as a function of velocity and time

Displacement as a function of time, velocity, and acceleration

Velocity as a function of Displacement and acceleration

You may use different forms of Kinetic equations, depending on the information given to you for specific problems!!


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Solving Problems (p.28)• Read problem carefully• Draw a diagram (choose coordinate system)• Write down known and unknown quantities• Think!• Calculate (substitute numbers at end)• Is answer reasonable?• Check units

Air Jordan revisited


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Coordinate Systems (Start Ch. 3)• Makes it easy to express locations or positions• Two commonly used systems, depending on convenience

– Cartesian (Rectangular) Coordinate System• Coordinates are expressed in (x,y)

– Polar Coordinate System • Coordinates are expressed in (r)

• Vectors become a lot easier to express and compute

O (0,0)

(x1,y1)=(r)r

sin

cos

ry

rx

How are Cartesian and Polar coordinates related?

1

1tan

21

21

x

y

yxr

y1

x1

+x

+y


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Example Cartesian Coordinate of a point in the xy plane are (x,y)= (-3.50,-2.50)m. Find the polar coordinates of this point.

y

x

(-3.50,-2.50)m

r

s

)(30.45.18

50.250.322

22

m

yxr

s 180

7

5

50.3

50.2tan

s

5.357

5tan 1

s

2165.35180180 s


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Vector and ScalarVector quantities have both magnitude (size) and direction

Scalar quantities have magnitude onlyCan be completely specified with a value and its unit

Force, acceleration, momentum

Normally denoted in BOLD BOLD letters, FF, or a letter with arrow on top FTheir sizes or magnitudes are denoted with normal letters, F, or absolute values: For F

Energy, heat, mass, speedNormally denoted in normal letters, E

Both have units!!!


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Properties of Vectors• Two vectors are the same if their and are the

same, no matter where they are on a coordinate system.

x

y

AABB

EE

DD

CC

FF

Which ones are the same vectors?

A=B=E=DA=B=E=D

Why aren’t the others?

C:C: The same magnitude but opposite direction: C=-A:C=-A:A negative vector

F:F: The same direction but different magnitude

sizes directions


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Vector Operations• Addition:

– Triangular Method: One can add vectors by connecting the tail of one vector to the tip of the other (tail-to-tip)

– Parallelogram method: Connect the tails of the two vectors and extend– Addition is commutative: Changing order of operation does not affect the results

A+B=B+AA+B=B+A, A+B+C+D+E=E+C+A+B+DA+B+C+D+E=E+C+A+B+D

AA

BBAA

BB=AA

BBA+BA+B

• Subtraction: – The same as adding a negative vector:A A - B = A B = A + (-BB)

AA-B-B

• Multiplication by a scalar is increasing the magnitude A, BA, B=2A A

AA B=2AB=2A

AB 2

A+BA+BA+BA+B

A-BA-B

OR


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Vector Components• Coordinate systems are useful for resolving vectors into their

components (lengths along coordinate axes…)• This makes vector algebra easier• Don’t forget about direction of vectors

22yx AAA

(Ax,Ay)AA

Ay

Ax

x

ycosAAx

AA

AAA

222

22

sincos

sincos

}Components(+,+)

(-,+)

(-,-) (+,-)

sinAAy

} Magnitude

Example onboard


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Unit Vectors• Unit vectors indicate the directions of the

components• Dimensionless • Magnitudes are exactly 1• Unit vectors are usually expressed as i, j, k for x, y,

and z axis, respectively orkji , ,

jAiAjAiAA yx sincos So the vector AA can be re-written as


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Examples of Vector Addition

mjiji 0.20.40.40.20.20.2

)(0.75925 cmkji

Find the resultant vector which is the sum of AA=(2.0ii+2.0jj) and B B =(2.0ii-4.0jj)

jijiBAC 0.40.20.20.2

Find the resultant displacement of three consecutive displacements: dd11=(15ii+30j j +12kk)cm, dd22=(23ii+14j j -5.0kk)cm, and dd11=(-13ii+15jj)cm

)(650.75925 222 cmD

)(5.4200.4160.20.4 22 mC

kji 0.512151430132315

321 dddD

1 1 2.0tan tan 27

4.0y

x

C

C

OR Magnitude

Direction

jikjikji 15130.51423123015

Magnitude

Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 1 PHYS 1443 – Section 501...

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