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Transcript Welcome to Physics 211!
Classical Mechanics
Lecture 2-Two-dimensional Kinematics
Today's Concepts:
a) Vectors
b) Projectile motion
c) Reference frames
Reminder Lectures are posted online @
http://www.physics.utah.edu/~springer/phys1500/lectures.html
Mechanics Lecture 2, Slide 1
Projectile Motion Example -Trebuchet
https://www.youtube.com/watch?v=JQLYHt-DM0Q
How far did it go?
How high did it go?
What was its launch velocity?
How high did it go?
Mechanics Lecture 2, Slide 2
Projectile Motion
Trebuchet
http://www.sciencebuddies.org/science-fairprojects/project_ideas/ApMech_p013.shtml
Catapault Projectile Motion
http://www.yale.edu/ynhti/curriculum/units/2012/4/12.0
4.03.x.html
Mechanics Lecture 2, Slide 3
Homework 1
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Mechanics Lecture 2, Slide 4
Smartphysics links
Link to earlier unit
Mechanics Lecture 2, Slide 5
Homework Frustration
I experienced some degree
of “frustration” with the
Homework Problems
A.
B.
True
False
Mechanics Lecture 1, Slide 6
Homework Accomplishment
I experienced a sense of
“accomplishment” solving
the Homework Problems
A.
B.
True
False
Mechanics Lecture 1, Slide 7
Need Vectors!!! to describe motion in 2 & 3 dimensions
Mechanics Lecture 2, Slide 8
Vectors and 2-d kinematics –Main Points
Mechanics Lecture 2, Slide 9
Vectors and 2-d kinematics –Main Points
Mechanics Lecture 2, Slide 10
Vectors and 2d-kinematics
Important Equations
Kinematics
Mechanics Lecture 2, Slide 11
Lecture Thoughts
Mechanics Lecture 2, Slide 12
Vectors
A
Vector : directed line segment
q
http://mathforum.org/~klotz/Vectors/VandP.1.html
Think of a vector as an arrow.
(An object having both magnitude and direction)
The object is the same no matter how we chose to describe it
Mechanics Lecture 2, Slide 13
Vectors-Cartesian Coordinates/Components
Cartesian
Magnitude (length)
Independent of coordinate system
A Ax2 Ay2
Ay
A
A
q
A cos q A sin q
Direction
Ax
A Ax , Ay
2
q
2
A cos 2 q sin 2 q A
Depends on coordinate system
Cartesian Components
Ax A cosq
Ay A sin q
The object is the same no matter how we chose to describe it
Mechanics Lecture 2, Slide 14
Vectors-Polar Coordinates
Polar
Magnitude (length)
Independent of coordinate system
A Ax2 Ay2
A
q
A
A cos q A sin q
Direction
A A ,q
2
q
2
A cos 2 q sin 2 q A
Depends on coordinate system
Polar Components
A
q
The object is the same no matter how we chose to describe it
Mechanics Lecture 2, Slide 15
Vectors
Mechanics Lecture 2, Slide 16
Vectors in 3D
A vector can be defined in 2 or 3 (or even more) dimensions:
Mechanics Lecture 2, Slide 17
Vectors
The object is the same no matter how we chose to describe it
Decompose vector along unit vectors!!!
Mechanics Lecture 2, Slide 18
Vector Addition
AddTail to Head
Add
Components!!!
Cx Ax Bx
C y Ay By
Mechanics Lecture 2, Slide 19
Clicker Question A+B
Vectors Aand Bare shown to the right.
Which of the following best describes A+ B
A
B
C
D
A.
B.
A
C.
D.
E.
B
E
0%
0%
0%
0%
0%
Mechanics Lecture 2, Slide 20
Clicker Question A-B
Vectors Aand Bare shown to the right.
Which of the following best describes A- B
A
B
C
D
A.
B.
C.
A
D.
E.
B
E
0%
0%
0%
0%
0%
Mechanics Lecture 2, Slide 21
Clicker Question A+2B
A
Vectors Aand Bare shown to the right.
Which of the following best describes A+ 2 B
A
B
C
D
B
E
Mechanics Lecture 2, Slide 22
Acceleration Vector
Mechanics Lecture 2, Slide 23
Acceleration Vector
Mechanics Lecture 2, Slide 24
Kinematics in 3D
Mechanics Lecture 2, Slide 25
Checkpoint 1
Mechanics Lecture 2, Slide 26
Projectile Motion
Horizontal
Vertical
Boring
Mechanics Lecture 2, Slide 27
Train Demo Clicker Question
A flatbed railroad car is moving along a track at constant velocity.
A passenger at the center of the car throws a ball straight up.
Neglecting air resistance, where will the ball land?
A) Forward of the center of the car
correct
B) At the center of the car
C) Backward of the center of the car
vtrain car
Ball and car start with same x position and x velocity,
Since a = 0 they always have same x position.
Mechanics Lecture 2, Slide 28
Moving Rail Car
A.
B.
C.
A flatbed railroad car is moving along a track at constant velocity.
A passenger at the center of the car throws a ball straight up.
Neglecting air resistance, where will the ball land?
A) Forward of the center of the car
correct
B) At the center of the car
C) Backward of the center of the car
0%
0%
0%
vtrain car
Ball and car start with same x position and x velocity,
Since ax = 0 they always have same x position.
Mechanics Lecture 2, Slide 29
vtrain car
Time spend in the air depends on the maximum height
Maximum height depends on the initial vertical velocity
Mechanics Lecture 2, Slide 30
Monkey troubles
A.
B.
C.
You are a vet trying to shoot a tranquilizer dart into a monkey
hanging from a branch in a distant tree. You know that the monkey is
very nervous, and will let go of the branch and start to fall as soon as
your gun goes off. In order to hit the monkey with the dart, where
should you point the gun before shooting?
A) Right at the monkey
B) Below the monkey
C) Above the monkey
0%
0%
0%
Mechanics Lecture 2, Slide 31
Shooting the Monkey…
Dart
x vo t
1
y - gt2
2
Monkey
x xo
1
y - gt2
2
Mechanics Lecture 2, Slide 32
Shooting the Monkey…
Still works even if you shoot upwards!
y = voy t - 1/2 g t 2
y = yo - 1/2 g t 2
Dart hits the
monkey
Mechanics Lecture 2, Slide 33
Projectile Motion & Frames of Reference
Mechanics Lecture 2, Slide 34
Checkpoint 2
Destroyer
Enemy 1
A) Enemy 1
B) Enemy 2
C) They are both hit at the same time
Enemy 2
53% of you had incorrect answer…
Let’s try again.
Mechanics Lecture 2, Slide 35
Checkpoint 2
A.
B.
C.
…Which enemy ship gets hit first?
A) Enemy 1 B) Enemy 2 C) Same
0%
Destroyer
Enemy 1
0%
0%
Enemy 2
B) The height of the shell fired at ship 2 is less, so ship 2 gets hit first.
Mechanics Lecture 2, Slide 36
Checkpoint 3
A destroyer fires two shells with different initial speeds at two different
enemy ships. The shells follow the trajectories shown. Which enemy
ship gets hit first?
Destroyer
Enemy 1
Enemy 2
26% of you had incorrect answer…
Let’s try again.
A) Enemy 1
B) Enemy 2
C) They are both hit at the same time
Mechanics Lecture 2, Slide 37
Checkpoint 3
A.
B.
C.
…Which enemy ship gets hit first?
A) Enemy 1 B) Enemy 2 C) Same
0%
Destroyer
Enemy 1
0%
0%
Enemy 2
C) they both achieve the same height so they remain in the air the same amount of
time
Mechanics Lecture 2, Slide 38
Range
Mechanics Lecture 2, Slide 39
Range
MAXIMUM range OCCURS AT 450
f (q ) sin(2q )
df (q )
2 cos(2q )
dq
df (q )
0 cos(2q ) 0
dq
2q 900
q 450
Mechanics Lecture 2, Slide 40
Trigonometric Identity for range equation
eiq - e - iq
sin q
2i
eiq e -iq
cosq
2
ei - e -i ei e -i ei ei ei e -i - e -i ei - e -i e -i
sin cos
2
i
2
4i
ei ( ) ei ( - ) - ei ( - ) - e -i ( )
sin cos
4i
1 e i ( ) - e -i ( ) e i ( - ) - e - i ( - )
sin cos
2
2i
2i
sin cos
q
1
sin( ) sin( - )
2
sin q cosq
1
sin(q q ) sin(q - q ) 1 sin(2q )
2
2
http://mathworld.wolfram.com/Cosine.html
http://mathworld.wolfram.com/Sine.html
Mechanics Lecture 2, Slide 41
Trigonometric Identities relating sum and products
List of trigonometric identities
sin( ) sin cos cos sin
q
sin(2q ) sin q cosq cosq sin q 2 sin q cosq
Mechanics Lecture 2, Slide 42
Question 2
Mechanics Lecture 2, Slide 43
Question 2
Mechanics Lecture 2, Slide 44
Field Goal Example
A field goal kicker can kick the ball 30 m/s at an angle of 30 degrees w.r.t. the
ground. If the crossbar of the goal post is 3m off the ground, from how far
away can he kick a field goal?
y
x
3m
D
y-direction
x-direction
voy = vo sin(30o) = 15 m/s
vox = vo cos(30o) = 26 m/s
y = yo + voyt + ½ at 2
D = xo + vox t + ½ at 2
3 m = 0 m + (15 m/s) t – ½ (9.8 m/s2) t 2
= 0 m + (26 m/s)(2.8 s) + 0 m/s2 (2.8 s )2
t = 2.8 s or t = 0.22 s.
= 72.8 m
Illini Kicks 70 yard Field Goal
Mechanics Lecture 2, Slide 45
Vectors and 2d-kinematics – Main Points
Mechanics Lecture 2, Slide 46
Vectors and 2d-kinematics
Important Equations
Mechanics Lecture 2, Slide 47
Hyperphysics-Trajectories
http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html
Mechanics Lecture 1, Slide 48
Hyperphysics-Trajectories
Mechanics Lecture 1, Slide 49
Hyperphysics-Trajectories
Mechanics Lecture 1, Slide 50
Pre-lecture 2 viewing times
Mechanics Lecture 2, Slide 51