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

http://www.physics.utah.edu/~springer/phys1500/lectures.html

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?

https://www.youtube.com/watch?v=JQLYHt-DM0Q

Projectile Motion


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


13 students with 100%

21 students > 90%

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Smartphysics links


Link to earlier unit

Homework Frustration

A. TrueB. False


6%

94%

I experienced some degree of “frustration” with the Homework Problems

Homework Accomplishment

A. TrueB. False


21%

79%

I experienced a sense of “accomplishment” solving the Homework Problems


Need Vectors!!! to describe motion in 2 & 3 dimensions


Vectors and 2-d kinematics –Main Points

Vectors and 2d-kinematics Important Equations


Lecture Thoughts



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



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



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


Vectors


A vector can be defined in 2 or 3 (or even more) dimensions:

Vectors in 3D



Vectors

Decompose vector along unit vectors!!!


Vector Addition

AddTail to Head

yyy

xxx

BAC

BAC

Add Components!!!

Clicker Question A+B A. B. C. D. E.


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.


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


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


Acceleration Vector


Kinematics in 3D


Checkpoint 1


Horizontal Vertical BoringBoring

Projectile Motion


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.


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


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.


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


Monkey

x xo

Dart

x vo t

2

2

1gty

Shooting the Monkey…

2

2

1gty


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


Projectile Motion & Frames of Reference


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


Checkpoint 2 A. B. C.


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.


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.


0% 0%0%


…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


Range


0

0

45

902

0)2cos(0)(

)2cos(2)(

)2sin()(

d

dfd

df

f

MAXIMUM range OCCURS AT 450

Trigonometric Identity for range equation


)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


List of trigonometric identities

cossin2sincoscossin)2sin(

sincoscossin)sin(

Question 2



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


Vectors and 2d-kinematics – Main PointsVectors and 2d-kinematics – Main Points

Vectors and 2d-kinematics Important Equations


Hyperphysics-Trajectories


http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html

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mechanics lecture 2, slide 1 classical mechanics lecture 2-two-dimensional kinematics today's...

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