1. This Week 6/21 Lecture – Chapter 3 6/22 Recitation – Spelunk
Problems: 3.3, 3.29, 3.31
6/23 Lab – Kinematics in 1-D
6/24 Lecture – Chapter 4
6/25 Recitation – Projectile Motion
Problems: 4.7, 4.18, 4.28, 4.59
6/21/04

2. Chapter 3
Vectors

3. Vectors Vectors have a magnitude and a direction
In 1 dimension we could specify direction with +/-
In 2 and 3 dimensions we need more
Columbus to Harrisburg 315 mi and 5o wrt East
Columbus to Nashville 315 mi and 235o wrt East
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4. Notational Convention
How do we know when we are talking about a vector quantity?
Arrow A
Boldface A
Underline A
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5. Representation of Vectors
Graphical
Polar
Cartesian
Magnitude Direction
x and y components
Use the representation most appropriate to the problem
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6. Adding Vectors (Graphically)
Vectors are added “Tail to Tip”
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7. Trigonometry Review Sin q = opp/hyp Cos q = adj/hyp Tan q = opp/adj
hypotenuse
opposite
side
adjacent side
Sin q = opp/hyp
Cos q = adj/hyp
Tan q = opp/adj
Mnemonic: Soh Cah Toa
Note: p radians = 180o
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8. Unit Vectors Carry the “direction” information Unit magnitude
Dimensionless
velocity vector
x component of velocity vector
unit vector in the x direction
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9. Converting from Polar to Cartesianq r
Given:
Magnitude: r
Direction: q
x component of the vector
y component of the vector
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10. Converting from Cartesian to Polar
Given rx, ry: q r
Pythagorean Theorem
Geometry
Caution: Quadrant Ambiguity!
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11. Example: Find the magnitude and direction of a vector given by:
v = (vx ,vy) = (-5,-3) m/s y x v
vy=-3m/s f
vx=-5m/s
Note that f is in the -x,-y quadrant, so
q = 59° +180° =239°
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12. Example: Given the following velocity, calculate the acceleration
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13. Vector Addition (Quantitative)
Given two vectors
Add the components to get the vector sum
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14. Vector Addition We can verify this equation graphically x y Ax Bx ByAy
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15. Example:
Find the vector sum of the following:
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16. Scalar multiplication
Can multiply a vector by a scalar:
Example:
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17. Properties of Vectors Addition Commutative Property
Associative Property
Distributive Property
Distributive Property
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18. Properties of Vectors Associative Property Commutative Property
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19. Multiplication by a scalar
Vector Properties
Multiplication by a scalar
If c is positive:
Just changes the length of the vector
If c is negative:
Length of vector changes
Direction changes by 180°
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20. Dot or Scalar Product
f is the angle between the vectors if you put their tails together B f A
Recall that cos(f) = cos(-f), so
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21. Dot Product: Physical Meaning
Measures “how much” one vector lies along another B f A
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22. Example: Find the angle f between vectors A and B: First, solve for f:
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23. Solution:
Plugging these into our equation for f:
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24. “Cross” or Vector Product
Another way to multiply vectors…
Magnitude only…
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25. Example: Find the cross product of A and B Only in the xy-plane
Perpendicular to the xy-plane
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26. Right Hand Rule
Point your fingers in the direction of the first vector
Curl them in the direction of the second vector
Your thumb now points in the direction of the cross product
Sanity check your answers!
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27. Multiplication of Unit Vectors
Can multiply by components, but it usually takes longer and is easier to make mistakes +
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28. Properties of the Cross Product
Cross product normal to the surface of the plane created by A and B
Antisymmetric
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29. Properties of Vector Products
Antisymmetry
Multiplication by a scalar
Distributive property
Distributive property
Triple Product
Triple Product
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30. Vector Multiplication
Dot Product
result is a scalar
projection of one vector onto the other
Cross Product
result is a vector
resultant vector is perpendicular to both vectors
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31. Where to Shoot? a) Aim at target b) Aim ahead of targetvc
a) Aim at target
b) Aim ahead of
target
c) Aim behind target
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32. Where to Shoot? If Vc=15 m/s, Va=50 m/s, what is θ? sinθ=Vc/Va
Vnet θ
Vnet = Vc + Va
If Vc=15 m/s, Va=50 m/s,
what is θ?
Vnet vc Va θ
sinθ=Vc/Va
=15/50=0.3
 θ=17.5°
What is Vnet?
Vnet = Va cosθ = 50 cos(17.5)°
= 47.7 m/s
If the arena diameter is 60m,
how long is arrow flight?
Radius is 30m, time of flight is:
Ta=R/Vnet=(30 m)/(47.7 m/s)=0.63s
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33. Example (Problem 3.11)
A woman walks 250 m in the direction 30 east of north, then 175 m directly east. Find
the magnitude of her final displacement,
the angle of her final displacement, and
the distance she walks.
Which is greater, that distance or the magnitude of her displacement?
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34. Chapter 4
Motion in 2-D and 3-D

35. Chapter 4: 2D and 3D Motion
Ideas from 1D kinematics can be carried over into 2D and 3D.
Our kinematic variables are now vectors.
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36. Simplest Quantity: Position r(t)
Position of particle is specified by r(t)
which is a vector
depending on time
Separate into components
r(t)=x(t) î + y(t) ĵ
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37. Displacement
In 1-D
In 2-D
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38. Velocity Just like 1-D: Average Velocity Instantaneous Velocity
y(t)
x(t)
Direction is tangent to the path
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39. Velocity
(4m, 3m)
(5m, 2m)
Example: Say an object moves from point 1 to point 2 in 1 s. Find the average velocity of the object:
Two 1-D Problems
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40. Acceleration Just like 1-D: Average Accelerationvi vf a
Instantaneous Acceleration
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41. Treat each dimension separately
In x-direction: x, vx = dx/dt, and ax = dvx/dt
In y-direction: y, vy = dy/dt, and ay = dvy/dt
In z-direction: z, vz = dz/dt, and az = dvz/dt
Like three one-dimensional problems
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42. a and v Follow From r(t) v a r
Note: r, a, and v don’t have to point in the same direction!
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43. Example: (Problem 4.2) The position vector for an electron is
r = (5.0 m)i – (3.0 m)j – (2.0 m)k
Find the magnitude of r
Sketch the vector on a right handed coord system ^ ^ ^
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44. Example: (Problem 4.9)
A particle moves so that its position (in meters) as a function of time (in seconds) is r = i + 4 t2 j + t k. Write expressions for
its velocity as a function of time and
its acceleration as a function of time.
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