Motion%20in%20Two%20Dimensions

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PHYSICS Principles and Problems
Chapter 6 Motion in Two Dimensions
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Motion in Two Dimensions
CHAPTER6
BIG IDEA

• You can use vectors and Newtons laws to describe
  projectile motion and circular motion.

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Table Of Contents
CHAPTER6
Section 6.1 Projectile Motion Section 6.2
Circular Motion Section 6.3 Relative Velocity
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Projectile Motion
SECTION6.1
MAIN IDEA A projectiles horizontal motion is
independent of its vertical motion.
Essential Questions

• How are the vertical and horizontal motions of a
  projectile related?
• What are the relationships between a projectiles
  height, time in the air, initial velocity and
  horizontal distance traveled?

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Projectile Motion
SECTION6.1

• Review Vocabulary
• Motion diagram a series of images showing the
  positions of a moving object taken at regular
  time intervals.

• New Vocabulary
• Projectile
• Trajectory

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Projectile Motion
SECTION6.1
Path of a Projectile

• If you observed the movement of a golf ball being
  hit from a tee, a frog hopping, or a free throw
  being shot with a basketball, you would notice
  that all of these objects move through the air
  along similar paths, as do baseballs, and arrows.
  
• Each path rises and then falls, always curving
  downward along a parabolic path.

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Projectile Motion
SECTION6.1
Path of a Projectile (cont.)

• An object shot through the air is called a
  projectile.

• A projectile can be a football or a drop of
  water.
• You can draw a free-body diagram of a launched
  projectile and identify all the forces that are
  acting on it.

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Projectile Motion
SECTION6.1
Path of a Projectile (cont.)

• No matter what the object is, after a projectile
  has been given an initial thrust, if you ignore
  air resistance, it moves through the air only
  under the force of gravity.
• The force of gravity is what causes the object to
  curve downward in a parabolic flight path. Its
  path through space is called its trajectory.

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Projectile Motion
SECTION6.1
Path of a Projectile (cont.)

• You can determine an objects trajectory if you
  know its initial velocity.
• Two types of projectile motion are horizontal and
  angled.

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Projectile Motion
SECTION6.1
Independence of Motion in Two Dimensions
Click image to view movie.
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Projectile Motion
SECTION6.1
Angled Launches

• When a projectile is launched at an angle, the
  initial velocity has a vertical component as well
  as a horizontal component.
• If the object is launched upward, like a ball
  tossed straight up in the air, it rises with
  slowing speed, reaches the top of its path, and
  descends with increasing speed.

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Projectile Motion
SECTION6.1
Angled Launches (cont.)

• The adjoining figure shows the separate vertical-
  and horizontal-motion diagrams for the trajectory
  of the ball.

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Projectile Motion
SECTION6.1
Angled Launches (cont.)

• At each point in the vertical direction, the
  velocity of the object as it is moving upward has
  the same magnitude as when it is moving downward.
• The only difference is that the directions of
  the two velocities are opposite.

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Projectile Motion
SECTION6.1
Angled Launches (cont.)

• The adjoining figure defines two quantities
  associated with a trajectory.

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Projectile Motion
SECTION6.1
Angled Launches (cont.)

• One is the maximum height, which is the height of
  the projectile when the vertical velocity is zero
  and the projectile has only its
  horizontal-velocity component.

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Projectile Motion
SECTION6.1
Angled Launches (cont.)

• The other quantity depicted is the range, R,
  which is the horizontal distance that the
  projectile travels when the initial and final
  heights are the same.
• Not shown is the flight time, which is how much
  time the projectile is in the air.

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Projectile Motion
SECTION6.1
Angled Launches (cont.)

• For football punts, flight time is often called
  hang time.

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Projectile Motion
SECTION6.1
The Flight of a Ball
A ball is launched at 4.5 m/s at 66 above the
horizontal. What are the maximum height and
flight time of the ball?
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Step 1 Analyze and Sketch the Problem

• Establish a coordinate system with the initial
  position of the ball at the origin.

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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)

• Show the positions of the ball at the beginning,
  at the maximum height, and at the end of the
  flight.

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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)

• Draw a motion diagram showing v, a, and Fnet.

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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Identify the known and unknown variables.
Known yi 0.0 m ?i 66 vi 4.5 m/s
ay -9.8 m/s2 Vy, max 0.0 m/s
Unknown ymax ? t ?
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Step 2 Solve for the Unknown
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Find the y-component of vi.
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Substitute vi 4.5 m/s, ?i 66
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)

• Use symmetry to find the y-component of vf
• vyf - vyi - 4.1 m/s

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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Solve for the maximum height.
Vy,max2 vyi2 2a(ymax yi) (0.0 m/s2) vyi
2a(ymax 0.0m/s) ymax - vyi / 2a
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Substitute vyi 4.1 m/s, a -9.8 m/s2
ymax (4.1 m/s)2 0.86m 2(-9.8
m/s2)
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Solve for the time to return to the launching
height.
Vyf vyi at
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Solve for t.
t vyf vyi a
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Substitute vyf - 4.1 m/s, vyi 4.1 m/s,
a - 9.80 m/s2
-4.1 m/s 4.1 m/s 0.84 s -9.8 m/s2
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
Step 3 Evaluate the Answer
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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)

• Are the units correct?
• Dimensional analysis verifies that the units are
  correct.
• Do the signs make sense?
• All of the signs should be positive.
• Are the magnitudes realistic?
• 0.84 s is fast, but an initial velocity of 4.5
  m/s makes this time reasonable.

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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
The steps covered were

• Step 1 Analyze and Sketch the Problem
• Establish a coordinate system with the initial
  position of the ball at the origin.
• Show the positions of the ball at the beginning,
  at the maximum height, and at the end of the
  flight.
• Draw a motion diagram showing v, a, and Fnet.

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Projectile Motion
SECTION6.1
The Flight of a Ball (cont.)
The steps covered were

• Step 2 Solve for the Unknown
• Find the y-component of vi.
• Find an expression for time.
• Solve for the maximum height.
• Solve for the time to return to the launching
  height.
• Step 3 Evaluate the Answer

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Projectile Motion
SECTION6.1
Forces from Air

• So far, air resistance has been ignored in the
  analysis of projectile motion.
• Forces from the air can significantly change the
  motion of an object.
• Ex. Flying a kite, parachute for a skydiver.

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Projectile Motion
SECTION6.1
Forces from Air (cont.)

• Picture water flowing from a hose
• If wind is blowing in the same direction as the
  waters initial movement, the horizontal distance
  the water travels increases.
• If wind is blowing in the opposite direction as
  the waters initial movement, the horizontal
  distance the water travels decreases.
• Even if the air is not moving, it can have a
  significant effect on some moving objects.
• Ex. Paper falling to the ground.

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Section Check
SECTION6.1

• A boy standing on a balcony drops one ball and
  throws another with an initial horizontal
  velocity of 3 m/s. Which of the following
  statements about the horizontal and vertical
  motions of the balls is correct? (Neglect air
  resistance.)

A. The balls fall with a constant vertical
velocity and a constant horizontal
acceleration. B. The balls fall with a constant
vertical velocity as well as a constant
horizontal velocity. C. The balls fall with a
constant vertical acceleration and a constant
horizontal velocity. D. The balls fall with a
constant vertical acceleration and an increasing
horizontal velocity.
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Section Check
SECTION6.1
Answer
Reason The vertical and horizontal motions of a
projectile are independent. The only force acting
on the two balls is the force of gravity. Because
it acts in the vertical direction, the balls
accelerates in the vertical direction. The
horizontal velocity remains constant throughout
the flight of the balls.
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Section Check
SECTION6.1

• Which of the following conditions is met when a
  projectile reaches its maximum height?

A. The vertical component of the velocity is
zero. B. The vertical component of the velocity
is maximum. C. The horizontal component of the
velocity is maximum. D. The acceleration in the
vertical direction is zero.
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Section Check
SECTION6.1
Answer
Reason The maximum height is the height at which
the object stops its upward motion and starts
falling down, i.e. when the vertical component of
the velocity becomes zero.
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Section Check
SECTION6.1

• Suppose you toss a ball up and catch it while
  riding in a bus. Why does the ball fall in your
  hands rather than falling at the place where you
  tossed it?

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Section Check
SECTION6.1
Answer

• Trajectory depends on the frame of reference.
• For an observer on the ground, when the bus is
  moving, your hand is also moving with the same
  velocity as the bus, i.e. the bus, your hand, and
  the ball will have the same horizontal velocity.
  Therefore, the ball will follow a trajectory and
  fall back into your hands.

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Circular Motion
SECTION6.2
MAIN IDEA An object in circular motion has an
acceleration toward the circles center due to an
unbalanced force toward the circles center.
Essential Questions

• Why is an object moving in a circle at a constant
  speed accelerating?
• How does centripetal acceleration depend upon the
  objects speed and the radius of the circle?
• What causes centripetal acceleration?

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Circular Motion
SECTION6.2

• Review Vocabulary
• Average velocity the change in position divided
  by the time during which the change occurred the
  slope of an objects position-time graph.

• New Vocabulary
• Uniform circular motion
• Centripetal acceleration
• Centripetal force

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Circular Motion
SECTION6.2
Describing Circular Motion
Click image to view movie.
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Circular Motion
SECTION6.2
Centripetal Acceleration

• The angle between position vectors r1 and r2 is
  the same as that between velocity vectors v1 and
  v2.
• Thus, ?r/r ?v/v. The equation does not change
  if both sides are divided by ?t.

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• However, v ?r/?t and a ?v/?t

• Substituting v ?r/?t in the left-hand side and
  a ?v/?t in the right-hand side gives the
  following equation

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• Solve the equation for acceleration and give it
  the special symbol ac, for centripetal
  acceleration.

• Centripetal acceleration always points to the
  center of the circle. Its magnitude is equal to
  the square of the speed, divided by the radius of
  motion.

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• One way of measuring the speed of an object
  moving in a circle is to measure its period, T,
  the time needed for the object to make one
  complete revolution.
• During this time, the object travels a distance
  equal to the circumference of the circle, 2pr.
  The objects speed, then, is represented by v
  2pr/T.

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• The acceleration of an object moving in a circle
  is always in the direction of the net force
  acting on it, there must be a net force toward
  the center of the circle. This force can be
  provided by any number of agents.
• When an Olympic hammer thrower swings the hammer,
  the force is the tension in the chain attached to
  the massive ball.

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• When an object moves in a circle, the net force
  toward the center of the circle is called the
  centripetal force.

• To analyze centripetal acceleration situations
  accurately, you must identify the agent of the
  force that causes the acceleration. Then you can
  apply Newtons second law for the component in
  the direction of the acceleration in the
  following way.

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• The net centripetal force on an object moving in
  a circle is equal to the objects mass times the
  centripetal acceleration.

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• When solving problems, it is useful to choose a
  coordinate system with one axis in the direction
  of the acceleration.
• For circular motion, the direction of the
  acceleration is always toward the center of the
  circle.

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Circular Motion
SECTION6.2
Centripetal Acceleration (cont.)

• Rather than labeling this axis x or y, call it c,
  for centripetal acceleration. The other axis is
  in the direction of the velocity, tangent to the
  circle. It is labeled tang for tangential.
• Centripetal force is just another name for the
  net force in the centripetal direction. It is the
  sum of all the real forces, those for which you
  can identify agents that act along the
  centripetal axis.

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Circular Motion
SECTION6.2
Centrifugal Force

• According to Newtons first law, you will
  continue moving with the same velocity unless
  there is a net force acting on you.

• The passenger in the car would continue to move
  straight ahead if it were not for the force of
  the car acting in the direction of the
  acceleration.

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Circular Motion
SECTION6.2
Centrifugal Force (cont.)

• The so-called centrifugal, or outward force, is a
  fictitious, nonexistent force.

• You feel as if you are being pushed only because
  you are accelerating relative to your
  surroundings. There is no real force because
  there is no agent exerting a force.

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Section Check
SECTION6.2

• Explain why an object moving in a circle at a
  constant speed is accelerating.

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Section Check
SECTION6.2
Answer

• Acceleration is the rate of change of velocity,
  the object is accelerating due to its constant
  change in the direction of its motion.

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Section Check
SECTION6.2

• What is the relationship between the magnitude of
  centripetal acceleration (ac) and an objects
  speed (v)?

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Section Check
SECTION6.2
Answer
Reason From the equation for centripetal
acceleration Centripetal acceleration always
points to the center of the circle. Its magnitude
is equal to the square of the speed divided by
the radius of the motion.
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Section Check
SECTION6.2

• What is the direction of the velocity vector of
  an accelerating object?

A. toward the center of the circle B. away from
the center of the circle C. along the circular
path D. tangent to the circular path
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Section Check
SECTION6.2
Answer
Reason While constantly changing, the velocity
vector for an object in uniform circular motion
is always tangent to the circle. Vectors are
never curved and therefore cannot be along a
circular path.
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Relative Velocity
SECTION6.3
MAIN IDEA An objects velocity depends on the
reference frame chosen.
Essential Questions

• What is relative velocity?
• How do you find the velocities of an object in
  different reference frames?

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Relative Velocity
SECTION6.3

• Review Vocabulary
• resultant a vector that results from the sum of
  two other vectors.

• New Vocabulary
• Reference frame

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Relative Velocity
SECTION6.3
Relative Motion in One Dimension

• Suppose you are in a school bus that is traveling
  at a velocity of 8
  m/s in a positive direction. You walk with a
  velocity of 1 m/s toward the front of the bus.
• If a friend of yours is standing on the side of
  the road watching the bus go by, how fast would
  your friend say that you are moving?

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Relative Velocity
SECTION6.3
Relative Motion in One Dimension (cont.)

• If the bus is traveling at 8 m/s, this means that
  the velocity of the bus is 8 m/s, as measured by
  your friend in a coordinate system fixed to the
  road.
• When you are standing still, your velocity
  relative to the road is also 8 m/s, but your
  velocity relative to the bus is zero.

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Relative Velocity
SECTION6.3
Relative Motion in One Dimension (cont.)

• In the previous example, your motion is viewed
  from different coordinate systems.
• A coordinate system from which motion is viewed
  is a reference frame.

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Relative Velocity
SECTION6.3
Relative Motion in One Dimension (cont.)

• Walking at 1m/s towards the front of the bus
  means your velocity is measured in the reference
  from of the bus.
• Your velocity in the roads reference frame is
  different
• You can rephrase the problem as follows given
  the velocity of the bus relative to the road and
  your velocity relative to the bus, what is your
  velocity relative to the road?

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Relative Velocity
SECTION6.3
Relative Motion in One Dimension (cont.)

• When a coordinate system is moving, two
  velocities are added if both motions are in the
  same direction, and one is subtracted from the
  other if the motions are in opposite directions.

• In the given figure, you will find that your
  velocity relative to the street is 9 m/s, the sum
  of 8 m/s and 1 m/s.

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Relative Velocity
SECTION6.3
Relative Motion in One Dimension (cont.)

• You can see that when the velocities are along
  the same line, simple addition or subtraction can
  be used to determine the relative velocity.

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Relative Velocity
SECTION6.3
Relative Motion in One Dimension (cont.)

• Mathematically, relative velocity is represented
  as vy/b vb/r vy/r.
• The more general form of this equation is
• Relative Velocity va/b vb/c va/c
• The relative velocity of object a to object c is
  the vector sum of object as velocity relative to
  object b and object bs velocity relative to
  object c.

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Relative Velocity
SECTION6.3
Relative Motion in Two Dimensions

• The method for adding relative velocities also
  applies to motion in two dimensions.
• As with one-dimensional motion, you first draw a
  vector diagram to describe the motion and then
  you solve the problem mathematically.

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Relative Velocity
SECTION6.3
Relative Motion in Two Dimensions (cont.)

• For example, airline pilots must take into
  account the planes speed relative to the air,
  and their direction of flight relative to the
  air. They also must consider the velocity of the
  wind at the altitude they are flying relative to
  the ground.

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Relative Velocity
SECTION6.3
Relative Motion in Two Dimensions (cont.)

• You can use the equations in the figure to solve
  problems for relative motion in two dimensions.
• Velocity of a reference frame moving relative to
  the ground is .
• Velocity of an object in the moving frame is
  .

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Relative Velocity
SECTION6.3
Relative Velocity of a Marble
Ana and Sandra are riding on a ferry boat that is
traveling east at a speed of 4.0 m/s. Sandra
rolls a marble with a velocity of 0.75 m/s north,
straight across the deck of the boat to Ana. What
is the velocity of the marble relative to the
water?
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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Step 1 Analyze and Sketch the Problem

• Establish a coordinate system.

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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)

• Draw vectors to represent the velocities of the
  boat relative to the water and the marble
  relative to the boat.

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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Identify known and unknown variables.
Known vb/w 4.0 m/s vm/b 0.75 m/s
Unknown vm/w ?
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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Step 2 Solve for the Unknown
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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Because the two velocities are at right angles,
use the Pythagorean theorem.
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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Substitute vb/w 4.0 m/s, vm/b 0.75 m/s
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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Find the angle of the marbles motion.
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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Substitute vb/w 4.0 m/s, vm/b 0.75 m/s
11 north of east
The marble is traveling 4.1 m/s at 11 north of
east.
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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
Step 3 Evaluate the Answer
88
Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)

• Are the units correct?
• Dimensional analysis verifies that the velocity
  is in m/s.
• Do the signs make sense?
• The signs should all be positive.
• Are the magnitudes realistic?
• The resulting velocity is of the same order of
  magnitude as the velocities given in the problem.

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Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
The steps covered were

• Step 1 Analyze and Sketch the Problem
• Establish a coordinate system.
• Draw vectors to represent the velocities of the
  boat relative to the water and the marble
  relative to the boat.

90
Relative Velocity
SECTION6.3
Relative Velocity of a Marble (cont.)
The steps covered were

• Step 2 Solve for the Unknown
• Use the Pythagorean theorem.
• Step 3 Evaluate the Answer

91
Section Check
SECTION6.3

• Steven is walking on the top level of a
  double-decker bus with a velocity of 2 m/s toward
  the rear end of the bus. The bus is moving with a
  velocity of 10 m/s. What is the velocity of
  Steven with respect to Anudja, who is sitting on
  the top level of the bus and to Mark, who is
  standing on the street?

A. The velocity of Steven with respect to Anudja
is 2 m/s and is 12 m/s with respect to
Mark. B. The velocity of Steven with respect to
Anudja is 2 m/s and is 8 m/s with respect to
Mark. C. The velocity of Steven with respect to
Anudja is 10 m/s and is 12 m/s with respect to
Mark. D. The velocity of Steven with respect to
Anudja is 10 m/s and is 8 m/s with respect to
Mark.
92
Section Check
SECTION6.3
Answer
Reason The velocity of Steven with respect to
Anudja is 2 m/s since Steven is moving with a
velocity of 2 m/s with respect to the bus, and
Anudja is at rest with respect to the bus. The
velocity of Steven with respect to Mark can be
understood with the help of the following vector
representation.
93
Section Check
SECTION6.3

• Which of the following formulas correctly relates
  the relative velocities of objects a, b, and c to
  each other?

A. va/b va/c vb/c B. va/b ? vb/c va/c
C. va/b vb/c va/c D. va/b ? va/c vb/c
94
Section Check
SECTION6.3
Answer
Reason The relative velocity equation is
va/b vb/c
va/c. The relative velocity of object a to
object c is the vector sum of object as velocity
relative to object b and object bs velocity
relative to object c.
95
Section Check
SECTION6.3

• An airplane flies due south at 100 km/hr relative
  to the air. Wind is blowing at 20 km/hr to the
  west relative to the ground. What is the planes
  speed with respect to the ground?

A. (100 20) km/hr B. (100 - 20) km/hr
96
Section Check
SECTION6.3
Answer
Reason Since the two velocities are at right
angles, we can apply the Pythagorean theorem. By
using relative velocity law, we can write
vp/a2 va/g2 vp/g2
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98
Motion in Two Dimensions
CHAPTER6
Resources
Physics Online Study Guide Chapter Assessment
Questions Standardized Test Practice
99
Projectile Motion
SECTION6.1
Study Guide

• The vertical and horizontal motions of a
  projectile are independent. When there is no air
  resistance, the horizontal motion component does
  not experience an acceleration and has constant
  velocity the vertical motion component of a
  projectile experiences a constant acceleration
  under these same conditions.

100
Projectile Motion
SECTION6.1
Study Guide

• The curved flight path a projectile follows is
  called a trajectory and is a parabola. The
  height, time of flight, initial velocity and
  horizontal distance of this path are related by
  the equations of motion. The horizontal distance
  a projectile travels before returning to its
  initial height depends on the acceleration due to
  gravity an on both components on the initial
  velocity.

101
Circular Motion
SECTION6.2
Study Guide

• An object moving in a circle at a constant speed
  has an acceleration toward the center of the
  circle because the direction of its velocity is
  constantly changing.
• Acceleration toward the center of the circle is
  called centripetal acceleration. It depends
  directly on the square of the objects speed and
  inversely on the radius of the circle.

102
Circular Motion
SECTION6.2
Study Guide

• A net force must be exerted by external agents
  toward the circles center to cause centripetal
  acceleration.

103
Relative Velocity
SECTION6.3
Study Guide

• A coordinate system from which you view motion is
  called a reference frame. Relative velocity is
  the velocity of an object observed in a
  different, moving reference frame.

• You can use vector addition to solve motion
  problems of an object in a moving reference
  frame.

104
Motion in Two Dimensions
CHAPTER6
Chapter Assessment

• What is the range of a projectile?

A. the total trajectory that the projectile
travels B. the vertical distance that the
projectile travels C. the horizontal distance
that the projectile travels D. twice the maximum
height of the projectile
105
Motion in Two Dimensions
CHAPTER6
Chapter Assessment
Reason When a projectile is launched at an
angle, the straight (horizontal) distance the
projectile travels is known as the range of the
projectile.
106
Motion in Two Dimensions
CHAPTER6
Chapter Assessment

• Define the flight time of a trajectory.

A. time taken by the projectile to reach the
maximum height B. the maximum height reached by
the projectile divided by the magnitude of the
vertical velocity C. the total time the
projectile was in the air D. half the total time
the projectile was in the air
107
Motion in Two Dimensions
CHAPTER6
Chapter Assessment
Reason The flight time of a trajectory is
defined as the total time the projectile was in
the air.
108
Motion in Two Dimensions
CHAPTER6
Chapter Assessment

• What is centripetal force?

Answer When an object moves in a circle, the
net force toward the center of the circle is
called centripetal force.
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Chapter Assessment

• Donna is traveling in a train due north at 30
  m/s. What is the magnitude of the velocity of
  Donna with respect to another train which is
  running due south at 30 m/s?

A. 30 m/s 30 m/s B. 30 m/s - 30 m/s C. 302 m/s
302 m/s D. 302 m/s - 302 m/s
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• Reason The magnitude of the velocity of Donna
  relative to the ground is 30 m/s and the
  magnitude of velocity of the other train relative
  to the ground is also 30 m/s.
• Now, with this speed, if the trains move in the
  same direction, then the relative speed of one
  train to another will be zero.

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• Reason In this case, since the two trains are
  moving in opposite directions, the relative speed
  (magnitude of the velocity) of Donna relative to
  another train is 30 m/s 30 m/s.

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• What is the relationship between the magnitude of
  centripetal acceleration and the radius of a
  circle?

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Reason Centripetal acceleration always points to
the center of the circle. Its magnitude is equal
to the square of the speed, divided by the radius
of motion. That is,
Therefore,
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Chapter Assessment

• Which of the following formulas can be used to
  calculate the period (T) of a rotating object if
  the centripetal acceleration (ac) and radius (r)
  are given?

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Reason We know that the centripetal
acceleration always points toward the center of
the circle. Its magnitude is equal to the square
of the speed divided by the radius of motion.
That is, ac v2/r. To measure the speed of an
object moving in a circle, we measure the period,
T, the time needed for the object to make one
complete revolution. That is, v 2pr/T, where
2pr is the circumference of the circle.
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Reason Substituting for v in the equation
v 2pr/T, we
get,
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Standardized Test Practice

• A 1.60-m tall girl throws a football at an angle
  of 41.0 from the horizontal and at an initial
  velocity of 9.40 m/s. How far away from the girl
  will it land?

A. 4.55 m B. 5.90 m C. 8.90 m D. 10.5 m
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Standardized Test Practice

• A dragonfly is sitting on a merry-go-round, 2.8 m
  from the center. If the tangential velocity of
  the ride is 0.89 m/s, what is the centripetal
  acceleration of the dragonfly?

A. 0.11 m/s2 B. 0.28 m/s2 C. 0.32 m/s2 D. 2.2 m/s2
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Standardized Test Practice

• The centripetal force on a 0.82-kg object on the
  end of a 2.0-m massless string being swung in a
  horizontal circle is 4.0 N. What is the
  tangential velocity of the object?

A. 2.8 m/s2 B. 3.1 m/s2 C. 4.9 m/s2 D. 9.8 m/s2
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Standardized Test Practice

• A 1000-kg car enters an 80-m radius curve at 20
  m/s. What centripetal force must be supplied by
  friction so the car does not skid?

A. 5.0 N B. 2.5102 N C. 5.0103 N D. 1.0103 N
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Standardized Test Practice

• A jogger on a riverside path sees a rowing team
  coming toward him. If the jogger is moving at 10
  km/h, and the boat is moving at 20 km/h, how
  quickly does the jogger approach the boat?

A. 3 m/s B. 8 m/s C. 40 m/s D. 100 m/s
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Test-Taking Tip

• Practice Under Testlike Conditions

Answer all of the questions in the time provided
without referring to your book. Did you complete
the test? Could you have made better use of your
time? What topics do you need to review?
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Chapter Resources
Relative Velocity

• Another example of combined relative velocities
  is the navigation of migrating neotropical
  songbirds.
• In addition to knowing in which direction to fly,
  a bird must account for its speed relative to the
  air and its direction relative to the ground.

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Relative Velocity

• If a bird tries to fly over the Gulf of Mexico
  into a headwind that is too strong, it will run
  out of energy before it reaches the other shore
  and will perish.
• Similarly, the bird must account for crosswinds
  or it will not reach its destination.

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Chapter Resources
The Flight of a Ball
A ball is launched at 4.5 m/s at 66 above the
horizontal. What are the maximum height and
flight time of the ball?
126
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Chapter Resources
Relative Velocity of a Marble
Ana and Sandra are riding on a ferry boat that is
traveling east at a speed of 4.0 m/s. Sandra
rolls a marble with a velocity of 0.75 m/s north,
straight across the deck of the boat to Ana. What
is the velocity of the marble relative to the
water?
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Chapter Resources
Trajectories of Two Softballs
128
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Chapter Resources
Motion Diagrams for Horizontal and Vertical
Motions
129
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Chapter Resources
Projectiles Launched at an Angle
130
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Chapter Resources
The Flight of a Ball
131
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Chapter Resources
A Player Kicking a Football
132
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Chapter Resources
The Displacement of an Object in Circular Motion
133
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Chapter Resources
Vectors at the Beginning and End of a Time
Interval
134
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Uniform Circular Motion
135
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Chapter Resources
A Nonexistent Force
136
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Chapter Resources
Calculating Relative Velocity
137
Motion in Two Dimensions
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Chapter Resources
The Planes Velocity Relative to the Ground
138
Motion in Two Dimensions
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Chapter Resources
Relative Velocity of a Marble
139
Motion in Two Dimensions
CHAPTER6
Chapter Resources
A Hammer Thrower Swings a Hammer
140
End of Custom Shows