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- Vectors and
- Two-Dimensional 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 “tip-to-tail”
- 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 y-axes
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- The x-component of a vector is the projection along the x-axis
- The y-component of a vector is the projection along the y-axis
- Then,
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- The previous equations are valid only if θ is measured with respect
to the x-axis
- 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 x-axis
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- Choose a coordinate system and sketch the vectors
- Find the x- and y-components of all the vectors
- Add all the x-components
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- Add all the y-components
- 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 y-directions of motion can be treated independently
- The x-direction is uniform motion
- The y-direction is free fall
- The initial velocity can be broken down into its x- and y-components
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- x-direction
- ax = 0
- x = vxot
- This is the only operative equation in the x-direction since there is
uniform velocity in that direction
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- y-direction
- 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 x-direction
- All the general rules of projectile motion apply
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- Follow the general rules for projectile motion
- Break the y-direction 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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