Chapter 4 Motion in Two and Three Dimensions

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Chapter 4 Motion in Two and Three Dimensions

• 4.1. What is Physics?      
• 4.2. Position and Displacement      
• 4.3. Average Velocity and Instantaneous
  Velocity      
• 4.4. Average Acceleration and Instantaneous
  Acceleration      
• 4.5. Projectile Motion      
• 4.6. Projectile Motion Analyzed      
• 4.7. Uniform Circular Motion      
• 4.8. Relative Motion in One Dimension      
• 4.9. Relative Motion in Two Dimensions

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What is Physics?
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Position and Displacement

• Position vector

Displacement
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EXAMPLE 1 Displacement
In Fig., the position vector for a particle is
initially at
and then later is

What is the particle's displacement from
to
?
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Problem 2
A rabbit runs across a parking lot on which a set
of coordinate axes has, strangely enough, been
drawn. The coordinates of the rabbits position
as functions of time t (second) are given by
At t15 s, what is the rabbits position vector
in unit-vector notation and in magnitude-angle
notation?
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Average and Instantaneous Velocity
Instantaneous velocity is
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Particles Path vs Velocity
Displacement
The velocity vector
The direction of the instantaneous velocity of a
particle is always tangent to the particles path
at the particles position.
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Problem 3
A rabbit runs across a parking lot on which a set
of coordinate axes has, strangely enough, been
drawn. The coordinates of the rabbits position
as functions of time t (second) are given by
At t15 s, what is the rabbits velocity vector
in unit-vector notation and in magnitude-angle
notation?
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Average and Instantaneous Acceleration
                                                      

                                                      

Average acceleration is
Instantaneous acceleration is
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Speed up or slow down

• If the velocity and acceleration components along
  a given axis have the same sign then they are in
  the same direction. In this case, the object will
  speed up.
• If the acceleration and velocity components have
  opposite signs, then they are in opposite
  directions. Under these conditions, the object
  will slow down.

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Problem 4
A rabbit runs across a parking lot on which a set
of coordinate axes has, strangely enough, been
drawn. The coordinates of the rabbits position
as functions of time t (second) are given by
At t15 s, what is the rabbits acceleration
vector in unit-vector notation and in
magnitude-angle notation?
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How to solve two-dimensional motion problem?

• One ball is released from rest at the same
  instant that another ball is shot horizontally to
  the right

The horizontal and vertical motions (at right
angles to each other) are independent, and the
path of such a motion can be found by combining
its horizontal and vertical position components.
By Galileo
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Projectile Motion

• A particle moves in a vertical plane with some
  initial velocity but its acceleration is always
  the free-fall acceleration g, which is downward.
  Such a particle is called a projectile and its
  motion is called projectile motion.

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Properties of Projectile Motion

• The Horizontal Motion
• no acceleration
• velocity vx remains unchanged from its initial
  value throughout the motion
• The horizontal range R is maximum for a launch
  angle of 45

    




                                                                                                              

• The vertical Motion
• Constant acceleration g
• velocity vy0 at the highest point.

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Check Your Understanding

• A projectile is fired into the air, and it
  follows the parabolic path shown in the drawing.
  There is no air resistance. At any instant, the
  projectile has a velocity v and an acceleration
  a. Which one or more of the drawings could not
  represent the directions for v and a at any point
  on the trajectory?

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Example 4  A Falling Care Package
Figure shows an airplane moving horizontally with
a constant velocity of 115 m/s at an altitude of
1050 m. The directions to the right and upward
have been chosen as the positive directions. The
plane releases a care package that falls to the
ground along a curved trajectory. Ignoring air
resistance, (a). determine the time required for
the package to hit the ground. (b) find the
speed of package B and the direction of the
velocity vector just before package B hits the
ground.
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Example 5  The Height of a Kickoff

• A placekicker kicks a football at an angle of
  ?40.0o above the horizontal axis, as Figure
  shows. The initial speed of the ball is
• (a) Ignore air resistance and find the maximum
  height H that the ball attains.
• (b) Determine the time of flight between kickoff
  and landing.
• (c). Calculate the range R of the projectile.

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UNIFORM CIRCULAR MOTION

• Uniform circular motion is the motion of an
  object traveling at a constant (uniform) speed on
  a circular path

                                                                                        
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Properties of UNIFORM CIRCULAR MOTION

• Period of the motion T is the time for a
  particle to go around a closed path exactly once
  has a special name.

• Average speed is

• This number of revolutions in a given time is
  known as the frequency, f.

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Example 6  A Tire-Balancing Machine

• The wheel of a car has a radius of r0.29 m
  and is being rotated at 830 revolutions per
  minute (rpm) on a tire-balancing machine.
  Determine the speed (in m/s) at which the outer
  edge of the wheel is moving.

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CENTRIPETAL ACCELERATION

• Magnitude The centripetal acceleration of an
  object moving with a speed v on a circular path
  of radius r has a magnitude ac given by

• Direction The centripetal acceleration vector
  always points toward the center of the circle and
  continually changes direction as the object
  moves.

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Check Your Understanding

• The car in the drawing is moving clockwise
  around a circular section of road at a constant
  speed. What are the directions of its velocity
  and acceleration at following positions? Specify
  your responses as north, east, south, or west.
• position 1
• position 2

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Example 7  The Effect of Radius on Centripetal
Acceleration

• The bobsled track at the 1994 Olympics in
  Lillehammer, Norway, contained turns with radii
  of 33 m and 24 m, as Figure illustrates. Find the
  centripetal acceleration at each turn for a speed
  of 34 m/s, a speed that was achieved in the
  two-man event. Express the answers as multiples
  of g9.8 m/s2.

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Relative Motion in One Dimension


                                                                                                             
The coordinate                   
The velocity                   
The acceleration                   
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Relative Motion in Two Dimension


                                                                                                             
The coordinate                   
The velocity                   


The acceleration                   
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Sample Problem

• In Fig. 4-23a, a plane moves due east while
  the pilot points the plane somewhat south of
  east, toward a steady wind that blows to the
  northeast. The plane has velocity
  relative to the wind, with an airspeed (speed
  relative to the wind) of 215 km/h, directed at
  angle ? south of east. The wind has velocity
  vpG relative to the ground with speed of 65.0
  km/h, directed 20.0 east of north. What is the
  magnitude of the velocity of the plane relative
  to the ground, and what is ??