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 Vectors and
 TwoDimensional Motion

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 When handwritten, use an arrow:
 When printed, will be in bold print:
A
 When dealing with just the magnitude of a vector in print, an italic
letter will be used: A

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 Equality of Two Vectors
 Two vectors are equal if they have the same magnitude and the same
direction
 Movement of vectors in a diagram
 Any vector can be moved parallel to itself without being affected

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 Negative Vectors
 Two vectors are negative if they have the same magnitude but are 180°
apart (opposite directions)
 Resultant Vector
 The resultant vector is the sum of a given set of vectors

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 When adding vectors, their directions must be taken into account
 Units must be the same
 Graphical Methods
 Algebraic Methods

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 Choose a scale
 Draw the first vector with the appropriate length and in the direction
specified, with respect to a coordinate system
 Draw the next vector with the appropriate length and in the direction
specified, with respect to a coordinate system whose origin is the end
of vector A and parallel to the coordinate system used for A

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 Continue drawing the vectors “tiptotail”
 The resultant is drawn from the origin of A to the end of the last
vector
 Measure the length of R and its angle
 Use the scale factor to convert length to actual magnitude

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 When you have many vectors, just keep repeating the process until all
are included
 The resultant is still drawn from the origin of the first vector to the
end of the last vector

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 When you have only two vectors, you may use the Parallelogram Method
 All vectors, including the resultant, are drawn from a common origin
 The remaining sides of the parallelogram are sketched to determine the
diagonal, R

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 Vectors obey the Commutative Law of Addition
 The order in which the vectors are added doesn’t affect the result

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 Special case of vector addition
 If A – B, then use A+(B)
 Continue with standard vector addition procedure

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 The result of the multiplication or division is a vector
 The magnitude of the vector is multiplied or divided by the scalar
 If the scalar is positive, the direction of the result is the same as of
the original vector
 If the scalar is negative, the direction of the result is opposite that
of the original vector

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 A component is a part
 It is useful to use rectangular components
 These are the projections of the vector along the x and yaxes

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 The xcomponent of a vector is the projection along the xaxis
 The ycomponent of a vector is the projection along the yaxis
 Then,

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 The previous equations are valid only if θ is measured with respect
to the xaxis
 The components can be positive or negative and will have the same units
as the original vector
 The components are the legs of the right triangle whose hypotenuse is A
 May still have to find θ with respect to the positive xaxis

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 Choose a coordinate system and sketch the vectors
 Find the x and ycomponents of all the vectors
 Add all the xcomponents

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 Add all the ycomponents
 Use the Pythagorean Theorem to find the magnitude of the Resultant:
 Use the inverse tangent function to find the direction of R:

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 Using + or – signs is not always sufficient to fully describe motion in
more than one dimension
 Vectors can be used to more fully describe motion
 Still interested in displacement, velocity, and acceleration

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 The position of an object is described by its position vector, r
 The displacement of the object is defined as the change in its position

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 The average velocity is the ratio of the displacement to the time
interval for the displacement
 The instantaneous velocity is the limit of the average velocity as Δt
approaches zero
 The direction of the instantaneous velocity is along a line that is
tangent to the path of the particle and in the direction of motion

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 The average acceleration is defined as the rate at which the velocity
changes
 The instantaneous acceleration is the limit of the average acceleration
as Δt approaches zero

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 The magnitude of the velocity (the speed) can change
 The direction of the velocity can change
 Even though the magnitude is constant
 Both the magnitude and the direction can change

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 An object may move in both the x and y directions simultaneously
 It moves in two dimensions
 The form of two dimensional motion we will deal with is called projectile
motion

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 We may ignore air friction
 We may ignore the rotation of the earth
 With these assumptions, an object in projectile motion will follow a
parabolic path

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 The x and ydirections of motion can be treated independently
 The xdirection is uniform motion
 The ydirection is free fall
 The initial velocity can be broken down into its x and ycomponents

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 xdirection
 a_{x }= 0
 x = v_{xo}t
 This is the only operative equation in the xdirection since there is
uniform velocity in that direction

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 ydirection
 free fall problem
 take the positive direction as upward
 uniformly accelerated motion, so the motion equations all hold

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 The velocity of the projectile at any point of its motion is the vector
sum of its x and y components at that point

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 An object may be fired horizontally
 The initial velocity is all in the xdirection
 v_{o} = v_{x} and v_{y} = 0
 All the general rules of projectile motion apply

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 Follow the general rules for projectile motion
 Break the ydirection into parts
 up and down
 symmetrical back to initial height and then the rest of the height

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 It may be useful to use a moving frame of reference instead of a
stationary one
 It is important to specify the frame of reference, since the motion may
be different in different frames of reference
 There are no specific equations to learn to solve relative velocity
problems

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 The pattern of subscripts can be useful in solving relative velocity
problems
 Write an equation for the velocity of interest in terms of the
velocities you know, matching the pattern of subscripts

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