Mechanics Lecture 2, Slide 1 Classical Mechanics Lecture 2-Two-dimensional Kinematics Today's...
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Transcript of Mechanics Lecture 2, Slide 1 Classical Mechanics Lecture 2-Two-dimensional Kinematics Today's...
Mechanics Lecture 2, Slide 1
Classical Mechanics Lecture 2-Two-dimensional
Kinematics
Today's Concepts:a) Vectorsb) Projectile motionc) Reference frames
Reminder Lectures are posted online @http://www.physics.utah.edu/~springer/phys1500/lectures.html
Projectile Motion Example -Trebuchet
Mechanics Lecture 2, Slide 2
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?
Projectile Motion
Mechanics Lecture 2, Slide 3
http://www.sciencebuddies.org/science-fair-projects/project_ideas/ApMech_p013.shtml
http://www.yale.edu/ynhti/curriculum/units/2012/4/12.04.03.x.html
Trebuchet Catapault Projectile Motion
Homework 1
Mechanics Lecture 2, Slide 4
13 students with 100%
21 students > 90%
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Smartphysics links
Mechanics Lecture 2, Slide 5
Link to earlier unit
Homework Frustration
A. TrueB. False
Mechanics Lecture 1, Slide 6
6%
94%
I experienced some degree of “frustration” with the Homework Problems
Homework Accomplishment
A. TrueB. False
Mechanics Lecture 1, Slide 7
21%
79%
I experienced a sense of “accomplishment” solving the Homework Problems
Mechanics Lecture 2, Slide 8
Need Vectors!!! to describe motion in 2 & 3 dimensions
Mechanics Lecture 2, Slide 9
Vectors and 2-d kinematics –Main Points
Mechanics Lecture 2, Slide 10
Vectors and 2-d kinematics –Main Points
Vectors and 2d-kinematics Important Equations
Mechanics Lecture 2, Slide 11
Lecture Thoughts
Mechanics Lecture 2, Slide 12
Mechanics Lecture 2, Slide 13
The object is the same no matter how we chose to describe it
Vectors
Think of a vector as an arrow. (An object having both magnitude and direction)
A
http://mathforum.org/~klotz/Vectors/VandP.1.html
Vector : directed line segment
Mechanics Lecture 2, Slide 14
The object is the same no matter how we chose to describe it
Ax
Ay
Vectors-Cartesian Coordinates/Components
A
CartesianCartesian
sin
cos
AA
AA
y
x
22yx AAA
Magnitude (length)
AAAAA
2222
sincossincos
Direction
Cartesian Components
yx AAA ,
Depends on coordinate system
Independent of coordinate system
Mechanics Lecture 2, Slide 15
The object is the same no matter how we chose to describe it
Vectors-Polar Coordinates
A
PolarPolar
22yx AAA
Magnitude (length)
AAAAA
2222
sincossincos
Direction
,AA
Polar Components
A
Depends on coordinate system
Independent of coordinate system
Mechanics Lecture 2, Slide 16
Vectors
Mechanics Lecture 2, Slide 17
A vector can be defined in 2 or 3 (or even more) dimensions:
Vectors in 3D
Mechanics Lecture 2, Slide 18
The object is the same no matter how we chose to describe it
Vectors
Decompose vector along unit vectors!!!
Mechanics Lecture 2, Slide 19
Vector Addition
AddTail to Head
yyy
xxx
BAC
BAC
Add Components!!!
Clicker Question A+B A. B. C. D. E.
Mechanics Lecture 2, Slide 20
0% 0% 0%0%0%
Vectors and are shown to the right. Which of the following best describes +
A
B
A
B
A
B
A B C D E
Clicker Question A-BA. B. C. D. E.
Mechanics Lecture 2, Slide 21
0% 0% 0%0%0%
Vectors and are shown to the right. Which of the following best describes
A
B
A
B
A
B
A B C D E
Mechanics Lecture 2, Slide 22
B
2
A B C D E
Vectors and are shown to the right. Which of the following best describes +
A
Clicker Question A+2B
B
AA
A
B
Mechanics Lecture 2, Slide 23
Acceleration Vector
Mechanics Lecture 2, Slide 24
Acceleration Vector
Mechanics Lecture 2, Slide 25
Kinematics in 3D
Mechanics Lecture 2, Slide 26
Checkpoint 1
Mechanics Lecture 2, Slide 27
Horizontal Vertical BoringBoring
Projectile Motion
Mechanics Lecture 2, Slide 28
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 carB) At the center of the carC) Backward of the center of the car
correct
Ball and car start with same x position and x velocity, Since a = 0 they always have same x position.
Train Demo Clicker Question
vtrain car
Moving Rail Car A. B. C.
Mechanics Lecture 2, Slide 29
0% 0%0%
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 carB) At the center of the carC) Backward of the center of the car
correct
Ball and car start with same x position and x velocity, Since ax = 0 they always have same x position.
vtrain car
Mechanics Lecture 2, Slide 30
Time spend in the air depends on the maximum height
Maximum height depends on the initial vertical velocity
vtrain car
Monkey troubles A. B. C.
Mechanics Lecture 2, Slide 31
0% 0%0%
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 monkeyB) Below the monkeyC) Above the monkey
Mechanics Lecture 2, Slide 32
Monkey
x xo
Dart
x vo t
2
2
1gty
Shooting the Monkey…
2
2
1gty
Mechanics Lecture 2, Slide 33
Shooting the Monkey…
y = voy t 1/2 g t 2
Still works even if you shoot upwards!y = yo 1/2 g t 2
Dart hits themonkey
Mechanics Lecture 2, Slide 34
Projectile Motion & Frames of Reference
Mechanics Lecture 2, Slide 35
Checkpoint 2
A) Enemy 1
B) Enemy 2
C) They are both hit at the same time
53% of you had incorrect answer…Let’s try again.
Enemy 1Destroyer Enemy 2
Enemy 1Destroyer Enemy 2
Checkpoint 2 A. B. C.
Mechanics Lecture 2, Slide 36
0% 0%0%
…Which enemy ship gets hit first?A) Enemy 1 B) Enemy 2 C) Same
B) The height of the shell fired at ship 2 is less, so ship 2 gets hit first.
Mechanics Lecture 2, Slide 37
Enemy 1Destroyer
Checkpoint 3A 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?
Enemy 2
A) Enemy 1
B) Enemy 2
C) They are both hit at the same time
26% of you had incorrect answer…Let’s try again.
Checkpoint 3 A. B. C.
Mechanics Lecture 2, Slide 38
0% 0%0%
Enemy 1Destroyer Enemy 2
…Which enemy ship gets hit first?A) Enemy 1 B) Enemy 2 C) Same
C) they both achieve the same height so they remain in the air the same amount of time
Range
Mechanics Lecture 2, Slide 39
Range
Mechanics Lecture 2, Slide 40
0
0
45
902
0)2cos(0)(
)2cos(2)(
)2sin()(
d
dfd
df
f
MAXIMUM range OCCURS AT 450
Trigonometric Identity for range equation
Mechanics Lecture 2, Slide 41
)2sin(2
1)sin()sin(
2
1cossin
)sin()sin(2
1cossin
222
1cossin
4cossin
422cossin
2cos
2sin
)()()()(
)()()()(
i
ee
i
ee
i
eeee
i
eeeeeeeeee
i
ee
ee
i
ee
iiii
iiii
iiiiiiiiiiii
ii
ii
http://mathworld.wolfram.com/Cosine.html http://mathworld.wolfram.com/Sine.html
Trigonometric Identities relating sum and products
Mechanics Lecture 2, Slide 42
List of trigonometric identities
cossin2sincoscossin)2sin(
sincoscossin)sin(
Question 2
Mechanics Lecture 2, Slide 43
Question 2
Mechanics Lecture 2, Slide 44
Mechanics Lecture 2, Slide 45
Field Goal ExampleA 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-direction
voy = vo sin(30o) = 15 m/s
y = yo + voyt + ½ at 2
3 m = 0 m + (15 m/s) t – ½ (9.8 m/s2) t 2
t = 2.8 s or t = 0.22 s.
x-direction
vox = vo cos(30o) = 26 m/s
D = xo + vox t + ½ at 2
= 0 m + (26 m/s)(2.8 s) + 0 m/s2 (2.8 s )2
= 72.8 m
D
3 m
y
x
Illini Kicks 70 yard Field Goal
Mechanics Lecture 2, Slide 46
Vectors and 2d-kinematics – Main PointsVectors and 2d-kinematics – Main Points
Vectors and 2d-kinematics Important Equations
Mechanics Lecture 2, Slide 47
Hyperphysics-Trajectories
Mechanics Lecture 1, Slide 48
http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html
Hyperphysics-Trajectories
Mechanics Lecture 1, Slide 49
Hyperphysics-Trajectories
Mechanics Lecture 1, Slide 50
Pre-lecture 2 viewing times
Mechanics Lecture 2, Slide 51