Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
1
PHYS 1443 – Section 501Lecture #4
Monday, Feb. 2, 2004Dr. Andrew Brandt
Motion in Two DimensionsVector Properties and OperationsMotion under constant acceleration
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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?)
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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!!
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
4
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
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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!!!
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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
Monday, February 2, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt
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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
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