Classical Mechanics
Lecture 2
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
Link to lectures on canvas pages as well…
Mechanics Lecture 2, Slide 1
Participation
Mechanics Lecture 2, Slide 2
Vectors and 2d-kinematics – Main Points
Mechanics Lecture 2, Slide 3
Vectors and 2d-kinematics – Main Points
Mechanics Lecture 2, Slide 4
Vectors and 2d-kinematics
Important Equations
Mechanics Lecture 2, Slide 5
Vectors

A
q
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 6
Vectors
Polar
Cartesian
Ay

A
Ax
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 7
Vectors
Mechanics Lecture 2, Slide 8
Vectors
The object is the same no matter how we chose to describe it
Mechanics Lecture 2, Slide 9
Vector Addition
Add
Components!!!
AddTail to Head
Mechanics Lecture 2, Slide 10
A.
Enter Question Text


Vectors Aand Bare shown to the right. 
Which of the following best describes A + B
A
B
C
D
B.

A
C.
D.
E.

B
E
0%
0%
0%
0%
0%
Mechanics Lecture 2, Slide 11
A.
B.
Enter Question Text


Vectors Aand Bare shown to the right. 
Which of the following best describes A - B
A
B
C
D
C.

A
D.
E.

B
E
0%
0%
0%
0%
0%
Mechanics Lecture 2, Slide 12
Clicker Question

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 13
Acceleration Vector
Mechanics Lecture 2, Slide 14
Acceleration Vector
Mechanics Lecture 2, Slide 15
Vectors in 3D
A vector can be defined in 2 or 3 (or even more) dimensions:
Mechanics Lecture 2, Slide 16
Kinematics in 3D
Mechanics Lecture 2, Slide 17
Checkpoint 1
Mechanics Lecture 2, Slide 18
Projectile Motion
Horizontal
Vertical
Boring
Mechanics Lecture 2, Slide 19
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 20
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 a = 0 they always have same x position.
Mechanics Lecture 2, Slide 21
vtrain car
Time spend in the air depends on the maximum height
Maximum height depends on the initial vertical velocity
Mechanics Lecture 2, Slide 22
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 23
Shooting the Monkey…
Dart
x = vo t
1
y = - gt2
2
Monkey
x = xo
1
y = - gt2
2
Mechanics Lecture 2, Slide 24
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 25
Projectile Motion & Frames of Reference
Mechanics Lecture 2, Slide 26
Checkpoint 2
Destroyer
Enemy 1
A) Enemy 1
B) Enemy 2
C) They are both hit at the same time
Enemy 2
60% of you had incorrect answer…
Let’s try again.
Mechanics Lecture 2, Slide 27
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 28
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
A) Enemy 1
B) Enemy 2
C) They are both hit at the same time
Enemy 2
66% of you had incorrect answer…
Let’s try again.
Mechanics Lecture 2, Slide 29
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 30
Range
Mechanics Lecture 2, Slide 31
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 32
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 33
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 34
Question 2
Mechanics Lecture 2, Slide 35
Question 2
Mechanics Lecture 2, Slide 36
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 37
Vectors and 2d-kinematics – Main Points
Mechanics Lecture 2, Slide 38
Vectors and 2d-kinematics
Important Equations
Mechanics Lecture 2, Slide 39
Hyperphysics-Trajectories
http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html
Mechanics Lecture 1, Slide 40
Hyperphysics-Trajectories
Mechanics Lecture 1, Slide 41
Hyperphysics-Trajectories
Mechanics Lecture 1, Slide 42