Unit 2: Motion in 2D

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Unit 2 Motion in 2D

• Textbook
• Chapter 3 Chapter 4

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Unit Objectives Motion Models

• Recognize that an object in free fall will
  accelerate at a constant rate of 9.8 m/s2
  downward near the surface of the earth.
• Use kinematic equations to determine velocity or
  position at any time
• Determine which model (constant velocity or
  constant acceleration, or varying acceleration)
  is appropriate to describe the horizontal and
  vertical component of motion of an object

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Unit Objectives Projectiles

• Use appropriate kinematic equations to determine
  the position or velocity of a projectile at a
  specific point. Sketch the graph of motion for
  projectiles
• a) y-x, y-t, x-t, vx t, vy- t, ax-t, ay-t
• Given information about the initial velocity and
  height of a projectile, determine a) time of
  flight, b) the point where a projectile lands,
  and c) velocity at impact

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Unit Objectives Vectors

• Graphical representation of Vectors
• Given a vector, draw its components
• Recognize the magnitude and direction of a vector
  from a vector diagram
• Determine the sum of 2 or more vectors
  graphically
• Numerical Analysis of Vectors
• Given the magnitude and direction of a vector,
  determine the components using trig
• Given the components of a vector, determine the
  magnitude and direction using Pythagorean Theorem
  and trig
• Determine the sum of 2 or more vectors using
  Pythagorean Theorem and trig
• Represent by using unit vectors i, j, k.

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Unit Objectives Relative Motion

• Use vectors to perform relative velocity
  calculations

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Free Fall Acceleration

• Acceleration due to the force of Earths gravity
• Acceleration due to gravity at the surface of the
  Earth is -9.8 m/s2. Negative because it points
  down.

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Graphs x-t v-t
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Free Fall on the Moon

• Acceleration of a falling object is constant
  regardless of mass or density

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Free Fall Key Points

• At max height, velocity is zero.
• At a given height, velocity up is equal to
  velocity down.
• Time up equals time down

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Vectors How much which way?

• When describing motion, often the questions asked
  are How far? or How fast?
• However, for a person that is lost, which way?
  becomes more valuable.

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Scalars vs Vectors

• Scalars have magnitude only
• Quantity of something
• Distance, speed, time, mass, temperature
• Vectors have both magnitude and direction
• displacement, velocity, acceleration

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Direction of Vectors

• The direction of a vector is represented by the
  direction in which the ray points.
• This is typically given by an angle.
• Can also be given by using unit vectors

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Magnitude of Vectors

• The magnitude of a vector is the size of whatever
  the vector represents.
• The magnitude is represented by the length of the
  vector.
• Symbolically, the magnitude is often represented
  as A

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Polar Notation

• Magnitude and direction of the vector are stated
  separately.
• Magnitude is a positive number and the angle is
  made with the positive x-axis

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Rectangular Notation

• Defining a vector by its components
• y-component vector projection parallel to y-axis
• x-component vector projection parallel to x-axis

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Converting Polar Rectangular
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Graphical Addition of Vectors

• Vectors are added graphically together
  head-to-tail.
• The sum is called the resultant.
• The inverse, is called the equilibrant .

A B R
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Component Addition of Vectors

• Resolve each vector into its x- and y-components.
• Ax Acos? Ay Asin?
• Bx Bcos? By Bsin?
• Add the x-components together to get Rx and the
  y-components to get Ry.
• Use the Pythagorean Theorem to get the magnitude
  of the resultant.
• Use the inverse tangent function to get the angle.

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Sample What is the value of a and b?
a -3 b 10
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Sample Problem

• Add together the following graphically and by
  component, giving the magnitude and direction of
  the resultant and of the equilibrant.
• Vector A 300 m _at_ 60o
• Vector B 450 m _at_ 100o
• Vector C 120 m _at_ -120o

Resultant 599 m _at_ 1o Equilibrant 599 m _at_ 181o
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Yet another sample!!!
Sprint (-6, -2) blocks
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Unit Vectors

• Unit vectors are quantities that specify
  direction only. They have a magnitude of exactly
  one, and typically point in the x, y, or z
  directions.

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Unit Vectors
z
k
j
y
i
x
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Unit Vectors

• Instead of using magnitudes and directions,
  vectors can be represented by their components
  combined with their unit vectors.
• Example displacement of 30 meters in the x
  direction added to a displacement of 60 meters in
  the y direction added to a displacement of 40
  meters in the z direction yields a displacement
  of

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Adding Vectors Using Unit Vectors

• Simply add all the i components together, all the
  j components together, and all the k components
  together.

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Sample Problem

• Consider two vectors, A 3.00 i 7.50 j and B
  -5.20 i 2.40 j. Calculate C where C A B.

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Sample Problem

• You move 10 meters north and 6 meters east. You
  then climb a 3 meter platform, and move 1 meter
  west on the platform. What is your displacement
  vector? (Assume East is in the x direction).

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Suppose I need to convert unit vectors to a
magnitude and direction?

• Given the vector

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Back to Sample Problem

• You move 10 meters north and 6 meters east. You
  then climb a 3 meter platform, and move 1 meter
  west on the platform. How far are you from your
  starting point?

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1 Dimension 2 or 3 Dimensions

• x position
• ?x displacement
• v velocity
• a acceleration

• r position
• ?r displacement
• v velocity
• a acceleration

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Sample Problem

• The position of a particle is given by r (80
  2t)i 40j - 5t2k. Derive the velocity and
  acceleration vectors for this particle. What does
  motion look like?

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Another Sample

• A position function has the form r x i y j
  with x t3 6 and y 5t - 3.
• What are the velocity and acceleration functions?
• What are the velocity and acceleration at t2s?

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Practice Problems

• 1- A baseball outfielder throws a long ball. The
  components of the position are x (30 t) m and y
  (10 t 4.9t2) m
• Write vector expressions for the balls position,
  velocity, and acceleration as functions of time.
  Use unit vector notation!
• Write vector expressions for the balls
  position, velocity, and acceleration at 2.0
  seconds.
• 2- A particle undergoing constant acceleration
  changes from a velocity of 4i 3j to a velocity
  of 5i j in 4.0 seconds. What is the
  acceleration of the particle during this time
  period? What is its displacement during this time
  period?

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Projectiles

• An object that moves in two dimensions under the
  influence of only gravity
• Accomplished by usually launching at an angle or
  going off a flat surface with initial horizontal
  velocity.
• Neglect air resistance
• Follow parabolic trajectory

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Launch Angle
The components vix viy are not necessarily
positive. If an object is thrown downward, then
viy is negative.
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Projectiles Acceleration

• If you take an object and drop, it will fall
  straight down and not sideways
• ax 0
• ayg -9.8 m/s2
• The vertical component of acceleration is just
  the familiar g of free fall while the horizontal
  is zero

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Trajectory of Projectile

• This shows the parabolic trajectory of a
  projectile fired over level ground.
• Acceleration points down at 9.8 m/s2 for the
  entire trajectory.

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Trajectory of Projectile
vx
vx
vy
vy
vx
vy
vx
vx
vy

• The velocity can be resolved into components all
  along its path. Horizontal velocity remains
  constant vertical velocity is accelerated.

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Trajectory Path of a Projectile
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Position graphs for 2-D projectiles. Assume
projectile fired over level ground.
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Acceleration graphs for 2-D projectiles. Assume
projectile fired over level ground.
ay
ax
t
t
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Lets think about this!!!

• A heavy ball is thrown exactly horizontally at
  height h above a horizontal field. At the exact
  instant that ball is thrown, a second ball is
  simply dropped from height h. Which ball hits
  firsts? (demo-x-y shooter)

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Two Independent Motions

• 1) Uniform motion at constant velocity in the
  horizontal direction
• 2) Free-Fall motion in the vertical direction

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RememberTo work projectile problems

• resolve the initial velocity into components.

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Practice Problems

• A soccer player kicks a ball at 15 m/s at an
  angle of 35o above the horizontal over level
  ground. How far horizontally will the ball travel
  until it strikes the ground?
• A cannon is fired at 100m/s at an 15o angle above
  the horizontal from the top of a 120 m high
  cliff. How long will it take the cannonball to
  strike the plane below the cliff? How far from
  the base of the cliff will it strike?
• Students at an engineering contest use a
  compressed air cannon to shoot a softball at a
  box being hoisted straight up at 10 m/s by a
  crane. The cannon, tilted upward at 30 degree
  angle, is 100 m from the box and fires by remote
  control the instant the box leaves the ground.
  Students can control the launch speed of the
  softball by setting air pressure. What launch
  speed should the students use to hit the box?

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Range Equation

• Derive the range equation for a projectile fired
  over level ground.

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Acceleration in 2-D

• A runner is going around a track. She is
  initially moving with a velocity vector of (0.00,
  -8.00) m/s and her constant acceleration is
  (1.10, 1.10) m/s2. What is her velocity 7.23
  seconds later. Round the final velocity
  components to the nearest 0.01 m/s.

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Multidimensional Motion - Calculus
What is the velocity function of the plane? What
is the velocity at t 2 seconds?
Just like in 1-D, take the derivative of the
position function, to get the velocity function.
Take the double derivative to find acelleration
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Unit Vectors Calculus

• Treat the same way as you do with one dimensional
  motion
• Take the derivative or integral for each unit
  vector

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Reference Frames

• Coordinate system used to make observations.

The woman is using the surface of the Earth as
her reference frame. She considers herself and
the train platform to be stationary, while the
train is moving to the right with positive
velocity.
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Reference Frames cont.
If now, the perception of motion is from Teds
point of view (man in the train). He uses the
inside of the train as his reference frame. He
sees other people in the train as stationary and
objects outside the train moving back with a
negative velocity.
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Reference Frames

• There is no right or wrong reference frame.
• Must be clear about which reference frame is
  being used to assess motion.

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Reference Frame Conditions

1. The frames are oriented the same, with the x and
   y axes parallel to each other
2. The origins of frame A B coincide at t0.
3. All motion is in the xy-plane, so we dont need
   to consider the z-axis
4. The relative velocity (of the frames) is
   constant. (a 0)

Inertial Reference Frames
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Inertial Reference Frames

• Classical Mechanics are only valid in inertial
  reference frames.
• In other words, all observers would measure the
  same acceleration for a moving body.
• We will discuss this in more detail when we talk
  about Newtons Laws of Motion

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Relative Velocity
v 15 m/s
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Another Sample
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Practice Problems
VBS 3.35 m/s at 63.4 degrees
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Police Car Chasing

• A motorist traveling west at 77.5 km/h is being
  chased by a police car traveling at 96.5 km/h.
  What is the velocity of the motorist relative to
  the police car?