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Chapter 3
Vectors and
Two-Dimensional Motion
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Vector vs. Scalar Review
All physical quantities encountered in this text will be either a scalar or a vector
A vector quantity has both magnitude (size) and direction
A scalar is completely specified by only a magnitude (size)
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Vector Notation
When handwritten, use an arrow:
When printed, will be in bold print with an arrow:
When dealing with just the magnitude of a vector in print, an italic letter will be used: A
A
A
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Properties of Vectors
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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More Properties of Vectors
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
; 0A B A A
R A B
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Adding Vectors
When adding vectors, their directions must be taken into account
Units must be the same
Geometric Methods Use scale drawings
Algebraic Methods More convenient
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Adding Vectors Geometrically (Triangle or Polygon Method)
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 and parallel to the coordinate system used for
AA
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Graphically Adding Vectors, cont.
Continue drawing the vectors “tip-to-tail”
The resultant is drawn from the origin of to the end of the last vector
Measure the length of and its angle Use the scale factor to
convert length to actual magnitude
A
R
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Graphically Adding Vectors, cont.
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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Notes about Vector Addition
Vectors obey the Commutative Law of Addition
The order in which the vectors are added doesn’t affect the result
A B B A
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Vector Subtraction
Special case of vector addition Add the negative
of the subtracted vector
Continue with standard vector addition procedure
A B A B
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Multiplying or Dividing a Vector by a Scalar
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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Components of a Vector
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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Components of a Vector, cont.
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,
cosA Ax
sinyA A
x yA A A
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More About Components of a Vector
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
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More About Components, cont.
The components are the legs of the right triangle whose hypotenuse is
May still have to find θ with respect to the positive x-axis
The value will be correct only if the angle lies in the first or fourth quadrant
In the second or third quadrant, add 180°
2 2 1tany
x y
x
AA A A and
A
A
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Adding Vectors Algebraically
Choose a coordinate system and sketch the vectors
Find the x- and y-components of all the vectors
Add all the x-components
This gives Rx:
xx vR
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Adding Vectors Algebraically, cont.
Add all the y-components
This gives Ry:
Use the Pythagorean Theorem to find the magnitude of the resultant:
Use the inverse tangent function to find the direction of R:
yy vR
2
y
2
x RRR
x
y1
R
Rtan
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Motion in Two Dimensions
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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Displacement
The position of an object is described by its position vector,
The displacement of the object is defined as the change in its position
r
f ir r r
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Velocity
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
avt
rv
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Acceleration
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
avt
va
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Unit Summary (SI)
Displacement m
Average velocity and instantaneous velocity m/s
Average acceleration and instantaneous acceleration m/s2
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Ways an Object Might Accelerate
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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Projectile Motion
Without gravity, a tossed object follows a straight-line path.
With gravity, the same object tossed at an angle follows a curved path.
Projectile
any object that moves through the air or space under the influence of gravity, continuing in motion by its own inertia
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Projectile Motion
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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Projectile Motion
Projectile motion is a combination of
a horizontal component
a vertical component
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Projectile Motion Projectiles launched horizontally
Important points:
horizontal component of velocity doesn’t change (when air drag is negligible) ball travels the same horizontal
distance in equal times (no
component of gravitational
force acting horizontally)
remains constant
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Projectile Motion vertical positions become farther apart with time
gravity acts downward, so ball accelerates downward
curvature of path is the combination of horizontal and vertical components of motion
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Projectile Motion Parabola
curved path of a projectile that undergoes acceleration only in the vertical direction, while moving horizontally at a constant speed
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Assumptions of Projectile Motion
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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Rules of Projectile Motion
The x- and y-directions of motion are completely independent of each other
The x-direction is uniform motion ax = 0
The y-direction is free fall ay = -g
The initial velocity can be broken down into its x- and y-components cos sinOx O O Oy O Ov v v v
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Projectile Motion
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Projectile Motion at Various Initial Angles
Complementary values of the initial angle result in the same range The heights will be
different
The maximum range occurs at a projection angle of 45o
Projectile trajectories
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Some Details About the Rules
x-direction
ax = 0
x = vxot
This is the only operative equation in the x-direction since there is uniform velocity in that direction
constantvcosvv xooxo
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More Details About the Rules
y-direction
free fall problem
a = -g
take the positive direction as upward
uniformly accelerated motion, so the motion equations all hold
ooyo sinvv
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Velocity of the Projectile
The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point
Remember to be careful about the
angle’s quadrant
2 2 1tany
x y
x
vv v v and
v
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Problem-Solving Strategy
Select a coordinate system and sketch the path of the projectile Include initial and final positions,
velocities, and accelerations
Resolve the initial velocity into x- and y-components
Treat the horizontal and vertical motions independently
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Problem-Solving Strategy, cont
Follow the techniques for solving problems with constant velocity to analyze the horizontal motion of the projectile
Follow the techniques for solving problems with constant acceleration to analyze the vertical motion of the projectile
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Some Variations of Projectile Motion
An object may be fired horizontally
The initial velocity is all in the x-direction vo = vx and vy = 0
All the general rules of projectile motion apply
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Non-Symmetrical Projectile Motion
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
Smashing Pumpkins
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Sample Problem
A fireman, 50.0 m away from a burning building, directs a stream of water from a fire hose at an angle of 30.0° above the horizontal. If the initial speed of the stream is 40.0 m/s, at what height will the stream of water strike the building?
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Fast-Moving Projectiles—Satellites
satellite motion is an example of a high-speed projectile
a satellite is simply a projectile that falls around Earth rather than into it
sufficient tangential velocity needed for orbit
with no resistance to reduce speed, a satellite goes around Earth indefinitely.
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As the ball leaves the girl’s hand, one second later it will have fallen 5 m
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Circular Satellite Orbits
Satellite in circular orbit
speed
must be great enough to ensure that its falling distance matches Earth’s curvature
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Circular Satellite Orbits
Curvature of the Earth
Earth surface drops a vertical distance of 5 meters for every 8000 meters tangent to the surface
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Circular Satellite Orbits
period for complete orbit about Earth
for satellites close to Earth—about 90 minutes
for satellites at higher altitudes—longer periods
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Camera on balloon at 100,000 feet
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Relative Velocity
Relative velocity is about relating the measurements of two different observers
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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Relative Velocity Notation
The pattern of subscripts can be useful in solving relative velocity problems
Assume the following notation:
E is an observer, stationary with respect to the earth
A and B are two moving cars
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Relative Position Equations
is the position of car A as measured by E
is the position of car B as measured by E
is the position of car A as measured by car B
AEr
ABr
BEr
AB AE EBr r r
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Relative Position
The position of car A relative to car B is given by the vector subtraction equation
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Relative Velocity Equations
The rate of change of the displacements gives the relationship for the velocities
AB AE EBv v v
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Problem-Solving Strategy: Relative Velocity
Label all the objects with a descriptive letter
Look for phrases such as “velocity of A relative to B”
Write the velocity variables with appropriate notation
If there is something not explicitly noted as being relative to something else, it is probably relative to the earth
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Problem-Solving Strategy: Relative Velocity, cont
Take the velocities and put them into an equation
Keep the subscripts in an order analogous to the standard equation
Solve for the unknown(s)
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Problem 3.50
A water insect maintains an average position on the surface of a stream by darting upstream (against the current) then drifting downstream (with the current) to its original position. The current in the stream is 0.500 m/s relative to the shore, and the insect darts upstream 0.560 m (relative to a spot on shore) in 0.800 s during the first part of its motion. Take upstream as the positive direction. (a) Determine the velocity of the insect relative to the water (i) during its dash upstream and (ii) during its drift downstream. (b) How far upstream relative to the water does the insect move during one cycle of its motion? (c) What is the average velocity of the insect relative to the water?