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
1. 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
2. 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
3. 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
4. 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
5. Graphical representation of Vectorsa) Given a vector, draw its components b) Recognize the magnitude and direction of a vector from
a vector diagram c) Determine the sum of 2 or more vectors graphically
6. Numerical Analysis of Vectors a) Given the magnitude and direction of a vector, determine
the components using trigb) Given the components of a vector, determine the
magnitude and direction using Pythagorean Theorem and trig
c) Determine the sum of 2 or more vectors using Pythagorean Theorem and trig
d) Represent by using unit vectors i, j, & k.
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Unit Objectives: Relative Motion
7. Use vectors to perform relative velocity calculations
Textbook:
Chapter 2: Section 26
Chapter 3 & Chapter 4
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Free Fall Acceleration
Acceleration due to the force of Earth’s 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
In 1971, the commander of Apollo 15 confirmed this concept by dropping a hammer and a feather. Both hit the ground at the same time.
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Free Fall – Key Points
1) At max height, velocity is zero.
2) At a given height, velocity up is equal to velocity down.
3) 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.
Vectors answer both questions:
1 – How much (magnitude)?
2 – Which way (direction)?
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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
R
headtail
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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
Ax
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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 │
AIf vector A represents a displacement of three kilometers to the north… B
Then vector B, which is twice as long, would represent a displacement of six kilometers to the north!
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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
v = 5 m/s at 135˚
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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
Vx
Vy
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Converting Polar & Rectangular
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A
B
R A + B = R
Graphical Addition of Vectors
Vectors are added graphically together head-to-tail. The sum is called the resultant. The inverse, is called the equilibrant .
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Component Addition of Vectors
1) Resolve each vector into its x- and y-components.Ax = Acos Ay = AsinBx = Bcos By = Bsin
2) Add the x-components together to get Rx and the y-components to get Ry.
3) Use the Pythagorean Theorem to get the magnitude of the resultant.
4) 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 @ 60o
Vector B: 450 m @ 100o
Vector C: 120 m @ -120o
Resultant: 599 m @ 1o
Equilibrant: 599 m @ 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.
ˆ points in the x direction
ˆ points in the y direction
ˆ points in the z direction
i
j
k
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Unit Vectors
z
y
x
i
jk
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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:
ˆˆ ˆ(30 -60 40 ) m
30,-60,40 m
i j k
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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.
C = -2.20 i + 9.90 j
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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).
5 i + 10 j + 3 k
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Suppose I need to convert unit vectors to a magnitude and direction?
Given the vector
2 2 2
ˆˆ ˆx y z
x y z
r r i r j r k
r r r r
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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?
11.56 m
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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
r = x i + y j + z k r = x i + y j + z k v = vx i + vy j + vz k
a = ax i + ay j + az k
In Unit VectorNotation
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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”?
v = 2 i - 10t k
a = -10 k
freefall
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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 t=2s?
v = 3t2 i + 5 j v(2) = 12 i + 5 j
a = 6t i a(2) = 12 i
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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 ball’s position, velocity, and
acceleration as functions of time. Use unit vector notation! Write vector expressions for the ball’s 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
cosiix vv
siniiy vv
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 &
ay=g = -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
g
g
g
g
g
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
vy
vy
vx
vx
vy
vx
vy
vx
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.
x
y
t
y
t
x
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Acceleration graphs for 2-D projectiles. Assume projectile fired over level ground.
t
ay
t
ax
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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
tvx x
atvv iyfy
2
2
1attvy iy
yavv iyfy 222
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Remember…To work projectile problems… …resolve the initial velocity into components.
VoVo,y = Vo sin
Vo,x = Vo cos
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Practice Problems
1) 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?
2) 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?
3. 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?
mx 57.21
smv /1.45
mx
st
88.286
97.2
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Range Equation
Derive the range equation for a projectile fired over level ground.
g
vR i 2sin2
tvx i )cos( cosiv
xt
cosiv
Rt
2cos
22
2
cossin0
iv
gR
iv
Riv
22
1 gttvy iy
2sincos220 gRRiv gRiv 2sin20
g
vR i 2sin2
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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.
smjiv f /)05.0,95.7(
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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 Ted’s 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 t=0.
3. All motion is in the xy-plane, so we don’t 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 Newton’s 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?
hkmvMG /5.77
hkmvPG /5.96
hkmvGP /5.96
vvv GPMGMP
hkmvMP /19