Kinematics in Two Dimensions

1
Kinematics in Two Dimensions
2
Section 1 Adding Vectors Graphically
3

• Adding Vectors Graphically
• Remember vectors have magnitude (length) and
  direction.
• When you add vectors you must maintain both
  magnitude and direction
• This information is represented by an arrow
  (vector)

4

• A vector has a magnitude and a direction
• The length of a drawn vector represents
  magnitude.
• The arrow represents the direction

Larger Vector
Smaller Vector
5
Graphical Representation of Vectors

• Given Vector a
• Draw 2a Draw -a

6
Problem set 1
a
c
b

1. Which vector has the largest magnitude?
2. What would -b look like?
3. What would 2 c look like?

7
Vectors
a
c
b

• Three vectors

8
a
c
b

• When adding vectors graphically, align the
  vectors head-to-tail.
• This means draw the vectors in order, matching up
  the point of one arrow with the end of the next,
  indicating the overall direction heading.
• Ex. a c
• The starting point is called the origin

c
a
origin
9
a
c
b

• When all of the vectors have been connected, draw
  one straight arrow from origin to finish. This
  arrow is called the resultant vector.

c
a
origin
10
a
c
b

• Ex.1 Draw a b

11
a
c
b

• Ex.1 Draw a b

Resultant
origin
12
a
c
b

• Ex. 2 Draw a b c

13
a
c
b

• Ex. 2 Draw a b c

Resultant
origin
14
a
c
b

• Ex. 3 Draw 2a b 2c

15
a
c
b

• Ex. 3 Draw 2a b 2c

origin
Resultant
16
Section 2 How do you name vector directions?
17
Vector Direction Naming

• How many degrees is this?

N
W
E
S
18
Vector Direction Naming

• How many degrees is this?

N
90º
W
E
S
19
Vector Direction Naming

• What is the difference between 15º North of East
  and 15 º East of North?

N
W
E
S
20
Vector Direction Naming

• What is the difference between 15º North of East
  and 15º East of North? (can you tell now?)

N
N
W
E
W
E
S
S
15º North of East
15º East of North
21
Vector Direction Naming
N
15º
W
S
15º North of what?
22
Vector Direction Naming
N
15º
W
E
S
15º North of East
23
15º
W
E
S
15º East of What?
24
N
15º
W
E
S
15º East of North
25
N E

• ___ of ___

This is the baseline. It is the direction you
look at first
This is the direction you go from the baseline to
draw your angle
26
Describing directions

• 30º North of East
• East first then 30º North
• 40º South of East
• East first then 30º South
• 25º North of West
• West first then 30º North
• 30º South of West
• West first then 30º South

27
Problem Set 2 (Name the angles)
30º
20º
45º
30º
20º
28
Intro Get out your notes
b

• Draw the resultant of
• a b c
• 2. What would you label following angles
• a. b.
• 3. Draw the direction 15º S of W

a
c
28º
18º
29
(No Transcript)
30
Section 3 How do you add vectors mathematically
(not projectile motion)
31
The Useful Right Triangle

• Sketch a right triangle and label its sides

c hypotenuse
a opposite
?
b adjacent
The angle
32

• The opposite (a) and adjacent (b) change based on
  the location of the angle in question
• The hypotenuse is always the longest side

?
c hypotenuse
b adjacent
a opposite
33

• The opposite (a) and adjacent (b) change based on
  the location of the angle in question
• The hypotenuse is always the longest side

?
c hypotenuse
b adjacent
a opposite
34
To figure out any side when given two other sides

• Use Pythagorean Theorem
• a2 b2 c2

c hypotenuse
a opposite
?
b adjacent
The angle
35
Sometimes you need to use trig functions
c hypotenuse
a opposite
?
a adjacent
Opp Hyp
Opp Adj
Sin ? _____
Tan ? _____
Adj Hyp
Cos ? _____
36
Sometimes you need to use trig functions
c hypotenuse
a opposite
?
a adjacent
Opp Hyp
Opp Adj
Sin ? _____
Tan ? _____
SOH CAH TOA
Adj Hyp
Cos ? _____
37
More used versions
Opp Hyp
Sin ? _____
Opp (Sin ?)(Hyp)
Adj Hyp
Cos ? _____
Adj (Cos ?)(Hyp)
Opp Adj
Opp Adj
? Tan-1 _____
Tan ? _____
38

• To resolve a vector means to break it down into
  its X and Y components.
• Example 85 m 25º N of W
• Start by drawing the angle

25º
39

• To resolve a vector means to break it down into
  its X and Y components.
• Example 85 m 25º N of W
• Start by drawing the angle
• The magnitude given is always the hypotenuse

85 m
25º
40

• To resolve a vector means to break it down into
  its X and Y components.
• Example 85 m 25º N of W
• this hypotenuse is made up of a X component
  (West)
• and a Y component (North)

85 m
North
25º
West
41

• In other words
• I can go so far west along the X axis and so far
  north along the Y axis and end up in the same
  place

finish
finish
85 m
North
origin
origin
25º
West
42

• If the question asks for the West component
  Solve for that side
• Here the west is the adjacent side
• Adj (Cos T)(Hyp)

85 m
25º
West or Adj.
43

• If the question asks for the West component
  Solve for that side
• Here the west is the adjacent side
• Adj (Cos T)(Hyp)
• Adj (Cos 25º)(85) 77 m W

85 m
25º
West or Adj.
44

• If the question asks for the North component
  Solve for that side
• Here the north is the opposite side
• Opp (Sin T)(Hyp)

85 m
North or Opp.
25º
45

• If the question asks for the North component
  Solve for that side
• Here the west is the opposite side
• Opp (Sin T)(Hyp)
• Opp (Sin 25º)(85) 36 m N

85 m
North or Opp
25º
46
Resolving Vectors Into Components

• Ex 4a. Find the west component of 45 m 19º S of W

47
Resolving Vectors Into Components

• Ex 4a. Find the west component of 45 m 19º S of W

48

• Ex 4a. Find the south component of 45 m 19º S of
  W

49

• Ex 4a. Find the south component of 45 m 19º S of
  W

50
Remember the wording. These vectors are at right
angles to each other.
5 m/s forward
5 m/s
Redraw and it becomes
30 m/s
Hypotenuse Resultant speed
velocity 30 m/s down
Right angle
51
Section 4 (Solving for a resultant)

• Ex. 6 Find the resultant of 35.0 m, N
• and 10.6 m, E.
• Start by drawing a vector diagram
• Then draw the resultant arrow

52

• Ex. 6 Find the resultant of 35.0 m, N
• and 10.6 m, E.
• Then draw the resultant vector and angle
• The angle you find is in the triangle closest to
  the origin

53

• Now we use Pythagorean theorem to figure out the
  resultant (hypotenuse)

 
54

• Then inverse tangent to figure out the angle
• The answer needs a magnitude, angle, and
  direction

 
 
 
 
55
 
 
 
 
 
56
Problem Set 3 Resolve the following vectors

• 48m, S and 25m, W
• 12.5m, S and 78m, N

57
Problem Set 3

• 48m, S and 25m, W

58
(No Transcript)
59
Section 4 How does projectile motion differ from
2D motion (without gravity)?
60
Projectile Motion
61

• Projectile- Object that is launched by a force
  and continues to move by its own inertia
• Trajectory- parabolic path of a projectile

62

• Projectile motion involves an object moving in 2D
  (horizontally and vertically) but only vertically
  is influenced by gravity.
• The X and Y components act independently from
  each other and will be separated in our
  calculations.

63
X and Y are independent

• X axis has uniform motion since gravity does not
  act upon it.

64
X and Y are Independent

• Y axis will be accelerated by gravity -9.8 m/s2

65
The equations for uniform acceleration, from unit
one, can be written for either x or y variables
66

• If we push the ball harder, giving it a greater
  horizontal velocity as it rolls off the table,
  the ball would take _________ time to fall to the
  floor.

67
Horizontal and vertical movement is independent

• If we push the ball harder, giving it a greater
  horizontal velocity as it rolls off the table,
  the ball would
• Y axis take the same time to fall to the floor.
  
• X axis It would just go further.

68
Solving Simple Projectile Motion Problems

• You will have only enough information to deal
  with the y or x axis first
• You cannot use the Pythagorean theorem since X
  and Y-axes are independent
• Time will be the key The time it took to fall is
  the same time the object traveled vertically.
• dx (vx)(t) is the equation for the
  horizontal uniform motion.
• If you dont have 2 of three x variable you will
  have to solve for t using gravity and the y axis

69
Equations Solving Simple Projectile Motion
Problems

• Do not mix up y and x variables
• dy height (this is negative if falling
  down)
• dx range (displacement x)

70
For all projectile motion problems

• Draw a diagram
• Separate the X and Y givens
• Something is falling in these problems
• X Givens Y Givens
• dX a -9.8 m/s
• vX
• t

71
Example Problem 8

• A stone is thrown horizontally at 7.50 m/s from a
  cliff that is 68.4 m high. How far from the base
  of the cliff does the stone land?

72
Write out your x and y givens separately

• A stone is thrown horizontally at 7.50 /s from a
  cliff that is 68.4 m high. How far from the base
  of the cliff does the stone land?

X givens
Y givens
73

• A stone is thrown horizontally at 7.50 m/s from a
  cliff that is 68.4 m high. How far from the base
  of the cliff does the stone land?

X givens
Y givens
74
Ex. 9

• A baseball is thrown horizontally with a velocity
  of 44 m/s. It travels a horizontal distance of
  18, to the plate before it is caught.
• How long does the ball stay in the air?
• How far does it drop during its flight?

75

• A baseball is thrown horizontally with a velocity
  of 44 m/s. It travels a horizontal distance of
  18, to the plate before it is caught.
• How long does the ball stay in the air?
• How far does it drop during its flight?

X givens
Y givens
76

• A baseball is thrown horizontally with a velocity
  of 44 m/s. It travels a horizontal distance of
  18, to the plate before it is caught.
• How long does the ball stay in the air?
• How far does it drop during its flight?

X givens
Y givens
77

• A baseball is thrown horizontally with a velocity
  of 44 m/s. It travels a horizontal distance of
  18, to the plate before it is caught.
• How long does the ball stay in the air?
• How far does it drop during its flight?

X givens
Y givens
78
Example10

• 1. What is the initial vertical velocity of the
  ball?

voY 0 m/s Same as if it was dropped from rest
79

• 2. How much time is required to get to the
  ground?

Since voY 0 m/s use
2(-10)
-10
t 1.4 s
80

• 3. What is the vertical acceleration of the ball
  at point A?

aoY -10 m/s2 always
81

• 4. What is the vertical acceleration at point B?

aoY -10 m/s2 always
82

• 5. What is the horizontal velocity of the ball at
  point C?

vX 5 m/s (does not change)
83

• 6. How far from the edge of the cliff does the
  ball land in the x plane?

X givens vX 5 m/s t 1.4 dx ?
dx (vX)(t) dx (5)(1.4) 7m
84

• What will happen if drops a package when the
  plane is directly over the target?

85

• The package has the same horizontal velocity as
  the plane and would land far away from the target.

86
Section 5 What do you do different if you have
projectile motion and V0Y is not equal to 0
87
Projectile Motion Concepts
Arrows represent x and y velocities (g always
10 m/s2 down)
88
Key points in a projectiles path
VoY 0 m/s

• When a projectile is at its highest point its vfy
  0. This means it stopped moving up.
• Use vfy 0 in a question that asks you to
  predict the vertical distance (how high)

89
Key points in a projectiles path

• If an object lands at the same height its
  vertical velocities final magnitude equals its
  initial but is in the opposite direction (down)

VoY 30 m/s
VfY -30 m/s
90

• The time it takes to rise to the top equals the
  time it takes to fall.
• Givens to use to find time to the top
• VoY 30 m/s VfY 0 m/s
• Givens to use to find time of entire flight
• VoY 30 m/s VfY -30 m/s

VoY 30 m/s
VfY -30 m/s
91
Key points in a projectiles path

• If a projectile lands below where it is launched
  the vfy magnitude will be greater than voy and in
  the reverse direction

92
Ex. 11 A ball of m 2kg is thrown from the
ground with a horizontal velocity of 5 m/s and
rises to a height of 45 m.

1. What happens to velocity in the x direction?
   Why?
2. What happens to velocity in the y direction? Why?

It stays constant during the entire flight (no
forces acting in the x direction)
It accelerates (the force of gravity is pulling
it to Earth)
93

• 3. Where is the projectile traveling the fastest?
  Why?
• 4. Where is the projectile traveling the slowest?
  What is its speed at this point?
• 5. Where is the acceleration of the projectile
  the greatest? Why?

A and E (has the largest VY component)
C (has only VX component VY0)
All (g stays -10m/s2)
94

• 6. What is the acceleration due to gravity at
  point B?
• 7. What is the initial vertical velocity the ball
  is thrown with?

All (g stays -10m/s2)
Must solve aY -10m/s2 d 45m vo ? Vf 0
vf2 vo2 2ad vo v(vf2 2ad)
vo v(02 2(-10)(45) vo 30 m/s up
95
VoY 30 m/s
VfY -30 m/s

• 8. What is the time required to reach point C if
  thrown from the ground?

Must solve
Y givens aY -10m/s2 vo 30 m/s Vf 0 m/s t
?
vf vo at t (vf vo) a
t (0 30) -10 t 3 s
96

• 9. From point C, what is the time needed to reach
  the ground?

Same as time it took to get to the top t 3 s
97

• 10. What is the horizontal velocity at point A?
• 11. What is the horizontal acceleration of the
  ball at point E?

5 m/s (never changes horizontally while in the
air)
ax 0 m/s2 (they asked for acceleration no
horizontal acceleration) vx stays 5 m/s
98

• 12. What is the vertical acceleration due to
  gravity at point E?

aY -10 m/s2
99

• 13. How far in the x plane (what is the range)
  does the ball travel?

Must solve
X givens t 6 seconds total in air vX 5 m/s dX
?
dX (vX)(t) dX (5)(6) 30 m
100

• 14. What would happen to the problem if the
  objects mass was 16 kg

Nothing would change. The acceleration due to
gravity is the same for any mass
101

• More complex projectile motion problems require
  you separate a resultant velocity vector into its
  components using soh-cah-toa
• A stone is thrown at 25 m/s at a 40º angle with
  the horizon. Start with the finding the vx and
  voy
• Then solve the problem like we have

voy
102
Example

• The punter on a football team tries to kick a
  football with an initial velocity of 25.0 m/s at
  an angle of 60.0º above the ground, what range
  (dx) does it travel?

103
Example

• The punter on a football team tries to kick a
  football with an initial velocity of 25.0 m/s at
  an angle of 60.0º above the ground, what range
  (dx) does it travel?

104
Example

• The punter on a football team tries to kick a
  football with an initial velocity of 25.0 m/s at
  an angle of 60.0º above the ground, what range
  (dx) does it travel?

105

• The punter on a football team tries to kick a
  football with an initial velocity of 25.0 m/s at
  an angle of 60.0º above the ground, what range
  (dx) does it travel?

106

• The punter on a football team tries to kick a
  football with an initial velocity of 25.0 m/s at
  an angle of 60.0º above the ground, what range
  (dx) does it travel?

107
45º will get you the greatest range

• Range is dx
• Horizontal displacement

108
Besides 45º, two sister angles will give you the
same range

• 45º is would give you the greatest dx
• Any similar degree variation on either side of
  45º would give you the same dx
• Ex these would give you the same dx.
• 40º and 50º
• 30º and 60º
• 15º would give you the same range as what?
  ___________

109
Classwork/Homework

• 2D motion Packet
• Pg 2 Exercise 10-16
• Honors Addition
• Book Pg 79 16,17,18,20,22,27,31
• Try 35