Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: INTRO TO MOTION IN 2D
● Motion at an angle (2D: A→C) is just combining TWO straight-line motions (1D: A→B & B→C), with vector equations.
- Whenever motion is in 2D, FIRST break it down into X & Y (1D).
MOTION EQs
+𝒚
𝒗𝒂𝒗𝒈
𝑪
𝒂𝒂𝒗𝒈
VECTOR EQs
𝚫𝒙
=
𝚫𝒕
𝚫𝒗
=
𝚫𝐭
𝐴⃗
𝜃𝑥
𝐴𝑥
UAM
(1) 𝒗 = 𝒗𝟎 + 𝒂𝒕
𝑨 = √𝑨𝒙𝟐 + 𝑨𝒚𝟐
(2) 𝒗𝟐 = 𝒗𝟎𝟐 + 𝟐𝒂𝚫𝒙
|𝑨𝒚 |
𝜽𝒙 = 𝐭𝐚𝐧−𝟏 (
)
|𝑨𝒙 |
𝟏
(3) 𝚫𝒙 = 𝒗𝟎 𝒕 + 𝒂𝒕𝟐
𝟐
𝑨
𝑩
+𝒙
𝐴𝑦
(4)* 𝚫𝒙 =
(𝒗𝟎 +𝒗)
𝟐
𝒕
𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝒙 )
𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝒙 )
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: POSITION AND DISPLACEMENT IN 2D
● If Position & Displacement are 2D vectors, use vector equations to jump between 2D vector ⇔ X & Y components.
ሬ⃗ ⇔ Components
Position _________ 𝒓
ሬ⃗ ⇔ Components
Displacement Vector 𝚫𝒓
Arrow from ___________ → Point
EXAMPLE 1: At point A, your position is 3.6m @ 33.7°.
You move to point B, where your position is 8.49m @ 45°.
Calculate the x & y components of your position at A & B.
+y
Shortest path from Point → Point
EXAMPLE 2: Using Example 1, calculate the magnitude &
direction of the displacement from A to B.
+y
B
A
A
+x
O
- Where you are / Coordinate (_____)
ሬ⃗| = ඥ𝒙𝟐 + 𝒚𝟐 𝒙
ሬ⃗ = 𝒓 cos θ
|𝒓
𝜽 = tan-1 ቀ
B
|𝒚|
|𝒙|
ቁ 𝒚
ሬ⃗ = 𝒓 sin θ
+x
O
- ______________ in position
ሬሬሬሬሬ⃗| = ඥ𝚫𝒙𝟐 + 𝚫𝒚𝟐
|𝚫𝒓
ሬ⃗ = 𝚫𝒓 cos θ
𝚫𝒙
|𝚫𝒚|
ቁ
|𝚫𝒙|
ሬ⃗ = 𝚫𝒓 sin θ
𝚫𝒚
𝜽 = tan-1 ቀ
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: At point A, a hiker is 10m east from the origin. After 35s, the hiker arrives at point B 40m at 60° north of east
from the origin. Calculate the magnitude and direction of the hiker’s displacement.
A)
B)
C)
D)
50m; 60° north of east
36m; 73.9° north of east
36m; 60° north of east
36m; 16.1°north of east
2D POSITION /
DISPLACEMENT EQUATIONS
ሬ⃗| = ඥ𝒙𝟐 + 𝒚𝟐 𝒙
ሬ⃗ = 𝒓 cos θ
|𝒓
𝜽 = tan-1 ቀ
|𝒚|
|𝒙|
ቁ 𝒚
ሬ⃗ = 𝒓 sin θ
ሬሬሬሬሬ⃗| = ඥ𝚫𝒙𝟐 + 𝚫𝒚𝟐
|𝚫𝒓
ሬ⃗ = 𝚫𝒓 cos θ
𝚫𝒙
|𝚫𝒚|
ቁ
|𝚫𝒙|
ሬ⃗ = 𝚫𝒓 sin θ
𝚫𝒚
𝜽 = tan-1 ቀ
PROBLEM: Your initial position is 6.2 m from the origin, 25° below the x-axis. You then travel 9.9 m at an angle 78° above
the positive x-axis, then 2.0 m in the negative x-direction. What is the magnitude & direction of your final position vector?
A)
B)
C)
D)
13.5m; 58° above +x-axis
18.1m; 78° above +x-axis
9.06m; 51.2° above +x-axis
10.4m; 42.6° above the x-axis
2D POSITION /
DISPLACEMENT EQUATIONS
ሬ⃗| = ඥ𝒙𝟐 + 𝒚𝟐 𝒙
ሬ⃗ = 𝒓 cos θ
|𝒓
𝜽 = tan-1 ቀ
|𝒚|
|𝒙|
ቁ 𝒚
ሬ⃗ = 𝒓 sin θ
ሬሬሬሬሬ⃗| = ඥ𝚫𝒙𝟐 + 𝚫𝒚𝟐
|𝚫𝒓
ሬ⃗ = 𝚫𝒓 cos θ
𝚫𝒙
|𝚫𝒚|
ቁ
|𝚫𝒙|
ሬ⃗ = 𝚫𝒓 sin θ
𝚫𝒚
𝜽 = tan-1 ቀ
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: SOLVING KINEMATICS PROBLEMS IN 2D
● Solving constant acceleration problems in 2D is done the same way as in 1D!
- Remember: Separate 2D motion into two 1D motions and solve.
EXAMPLE: A hockey puck slides along a lake at 8m/s east. A strong wind accelerates the puck at a constant 3 m/s2 in a
direction 37° northeast. What is the magnitude & direction of the hockey puck’s displacement after 5s?
2D MOTION w/ ACCELERATION
1) Draw Diagram & decompose vectors into x & y
2) List 5 variables for x & y, identify known &
target variables
3) Pick UAM Eq. without “Ignored” Variable
4) Solve
UAM Equations
X
Y
(1) 𝒗𝒙 = 𝒗𝟎𝒙 + 𝒂𝒙 𝒕
(1) 𝒗𝒚 = 𝒗𝟎𝒚 + 𝒂𝒚 𝒕
(2) 𝒗𝒙𝟐 = 𝒗𝟎𝒙𝟐 + 𝟐𝒂𝒙 𝜟𝒙
(2) 𝒗𝒚𝟐 = 𝒗𝟎𝒚𝟐 + 𝟐𝒂𝒚 𝜟𝒚
𝟏
(3) 𝜟𝒙 = 𝒗𝟎𝒙 𝒕 + 𝒂𝒙 𝒕𝟐
𝟐
(4) 𝜟𝒙 = (
𝒗𝒙 +𝒗𝟎𝒙
𝟐
)𝒕
𝟏
(3) 𝜟𝒚 = 𝒗𝟎𝒚 𝒕 + 𝒂𝒚 𝒕𝟐
𝟐
(4) 𝜟𝒚 = (
𝒗𝒚 +𝒗𝟎𝒚
𝟐
)𝒕
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: A survey drone has just completed a scan at x,y coordinates (57m, 8m) at t=0. It needs to return to a lab
located at (-115, 72) m. If its initial velocity is 16m/s in the +y-direction, and it has only 18s of battery life remaining, what
constant acceleration (magnitude and direction) does it need to reach the lab?
A)
B)
C)
D)
2.8 m/s2; along –x axis
1.8 m/s2; 51.8° below –x axis
2.7 m/s2; above –x axis
1.3 m/s2; 24° above –x axis
2D MOTION w/ ACCELERATION
1) Draw Diagram & decompose vectors into x & y
2) List 5 variables for x & y, identify known &
target variables
3) Pick UAM Eq. without “Ignored” Variable
4) Solve
MOTION EQs
𝒗𝒂𝒗𝒈
𝒂𝒂𝒗𝒈
VECTOR EQs
𝚫𝒙
=
𝚫𝒕
𝚫𝒗
=
𝚫𝐭
𝐴⃗
𝜃𝑥
𝐴𝑥
𝐴𝑦
UAM
(1) 𝒗 = 𝒗𝟎 + 𝒂𝒕
𝑨 = √𝑨𝒙𝟐 + 𝑨𝒚𝟐
(2) 𝒗𝟐 = 𝒗𝟎𝟐 + 𝟐𝒂𝚫𝒙
|𝑨𝒚 |
𝜽𝒙 = 𝐭𝐚𝐧−𝟏 (
)
|𝑨𝒙 |
𝟏
(3) 𝚫𝒙 = 𝒗𝟎 𝒕 + 𝒂𝒕𝟐
𝟐
(4)* 𝚫𝒙 =
(𝒗𝟎 +𝒗)
𝟐
𝒕
𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝒙 )
𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝒙 )
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: AVERAGE SPEED AND VELOCITY IN 2D
● Remember: Average speed & velocity measure how FAST something moves between two points.
Speed
Velocity
(Magnitude only)
(Magnitude + Direction)
𝒔=
𝐝𝐢𝐬𝐭𝐚𝐧𝐜𝐞
𝐭𝐢𝐦𝐞
𝒅
⇒ 𝚫𝒕
⃗ 𝒂𝒗𝒈 | =
|𝒗
- [ SCALAR | VECTOR ]
𝐝𝐢𝐬𝐩𝐥𝐚𝐜𝐞𝐦𝐞𝐧𝐭
𝐭𝐢𝐦𝐞
⇒
𝚫𝒕
𝜽𝒗 = ______
- [ SCALAR | VECTOR ]; always points in same direction as ____
EXAMPLE: You walk 40m in the +x-axis, then 30m in the +y-axis. The entire trip takes 10 seconds. Calculate
a) your average speed
b) the magnitude & direction of your velocity
+y
+x
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: While following a treasure map, you start at an old oak tree. You first walk 85 m at 30.0° west of north, then
walk 92 m at 67.0° north of east. You reach the treasure 2 minutes later. Calculate the magnitude of your average velocity
for the entire trip.
2D SPEED / VELOCITY
EQUATIONS
A) 1.11 m/s
B) 1.48 m/s
𝚫𝐫
𝐝
⃗ 𝒂𝒗𝒈 | ⇒
|𝒗
𝑠 ⇒ 𝚫𝒕
𝚫𝒕
C) 1.40 m/s
D) 1.32 m/s
PROBLEM: While following a treasure map, you start at an old oak tree. You first walk 85 m at 30.0° west of north, then
walk 92 m at 67.0° north of east. You reach the treasure 2 minutes later. Calculate your average speed for the entire trip.
A) 1.5 m/s
B) 177m/s
C) 88.5 m/s
2D SPEED / VELOCITY
EQUATIONS
𝐝
𝑠 ⇒ 𝚫𝒕
𝚫𝐫
⃗ 𝒂𝒗𝒈 | ⇒
|𝒗
𝚫𝒕
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: CALCULATING VELOCITY COMPONENTS
⃗ 𝒂𝒗𝒈 is 2D, it has x & y components. There are 2 sets of equations to go back & forth between 𝒗
⃗ 𝒂𝒗𝒈 & components:
● If 𝒗
1) Velocity Components ↔ Displacement & Time
2) Velocity Components ↔ Magnitude & Direction
EXAMPLE: You walk 40 m right, then 30 m up in 10s. Calculate the
velocity’s magnitude and its x & y components.
EXAMPLE: You walk at 5m/s at an angle 37° above the x-axis.
Calculate the x & y components of your velocity.
⃗ =5
𝒗
⃗|=
|𝒗
𝚫𝒚 = 30
𝚫𝒓
𝚫𝒕
=ξ
|
𝜽𝒗 = tan-1 ቀ|
⃗⃗⃗⃗
𝒗𝒙 = ______ = ___ cos θ
|
ቁ
|
𝒗𝒚 = ______= ___ sin θ
⃗⃗⃗
𝜽 = 37°
𝚫𝒙 = 40
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: A coastal breeze pushes your sailboat at constant velocity for 8 min. After checking your instruments, you
determine you’ve been pushed 650 m west and 800 m south. What was the magnitude & direction of your average velocity?
A) 2.15 m/s; 39.1° south of west
B) 128.9 m/s; 50.9° south of west
C) 2.15 m/s; 50.9° south of west
2D Velocity Vector
⃗|=
|𝒗
𝚫𝒓
𝚫𝒕
= ඥ𝑣𝑥2 + 𝑣𝑦2
|𝐯𝐲 |
𝜽𝒗 = tan-1 ൬|
𝐯𝐱 |
൰
⃗⃗⃗⃗
𝒗𝒙 =
𝚫𝒙
𝚫𝒕
= 𝒗 cos θ
𝚫𝒚
𝒗𝒚 =
⃗⃗⃗
= 𝒗 sin θ
𝚫𝒕
PROBLEM: A ball moves on a tabletop. The ball has initial x & y coordinates (1.8m, 3.6m). The ball moves 10m/s at 53.1°
above the x-axis for 4s. What are the x & y coordinates of the ball’s final position?
A)
B)
C)
D)
(11.8m,13.6m)
(25.8m, 35.6m)
(41.8m, 43.6m)
(33.8m, 27.6m)
2D Velocity Vector
⃗|=
|𝒗
𝚫𝒓
𝚫𝒕
= ඥ𝑣𝑥2 + 𝑣𝑦2
|𝐯𝐲 |
𝜽𝒗 = tan-1 ൬|
𝐯𝐱 |
൰
⃗⃗⃗⃗
𝒗𝒙 =
𝚫𝒙
𝚫𝒕
= 𝒗 cos θ
𝚫𝒚
𝒗𝒚 =
⃗⃗⃗
= 𝒗 sin θ
𝚫𝒕
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: ACCELERATION IN 2D
● Remember! Acceleration (in 1D & 2D) causes a change in _____________ (magnitude and/or direction).
- Just like velocity, there are two sets of equations to calculate acceleration and its components:
Acceleration
Velocity
ሬԦ
𝒗
|𝒗| =
𝚫𝒓
𝚫𝒕
= ට𝒗𝒙𝟐 + 𝒗𝒚𝟐
ሬԦ
𝒂
𝐯𝒚
𝐯𝒙
𝜽𝒗 = tan-1 ቀቚ ቚቁ
𝜃𝑣
𝒗𝒙 =
𝒗𝒚 =
𝚫𝒙
𝚫𝒕
𝚫𝒚
𝚫𝒕
𝚫𝒗
𝚫𝒕
=
ξ
𝜽𝒂 = tan-1 ቀቚ
𝒂𝒙 =
= 𝒗 𝒄𝒐𝒔𝜽
𝜃𝑎
= 𝒗 𝒔𝒊𝒏𝜽
|𝒂| =
𝒂𝒚 =
𝚫𝒕
𝚫𝒕
ቚቁ
= ____𝒄𝒐𝒔𝜽
= ____𝒔𝒊𝒏𝜽
EXAMPLE: A toy car is initially moving 20m/s in the +x-axis. 10 seconds later, the car is moving 67 m/s at 26.5° above the
x-axis. a) Calculate the x & y components of the car’s acceleration. b) Calculate the magnitude & direction of the car’s
acceleration over the 10s.
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: A football at rest is kicked by a football kicker. The ball is in contact with the kicker’s foot for 0.050s, during
which it experiences an acceleration 𝑎 = 340 m/s2. The ball is launched at an angle of 40° above the ground (x-axis).
Calculate the horizontal and vertical components of the launch velocity.
A)
B)
C)
D)
13 m/s horizontal; 10.9 m/s vertical
130 m/s horizontal; 109 m/s vertical
10.9 m/s horizontal; 13 m/s vertical
17 m/s horizontal; 17 m/s vertical
2D Acceleration Vector
ሬԦ| =
|𝒂
𝚫𝒗
𝚫𝒕
= ඥ𝑎𝑥2 + 𝑎𝑦2
ห𝐚𝐲 ห
𝜽𝒂 = tan-1 ൬|
𝐚𝐱 |
൰
ሬሬሬሬԦ
𝒂𝒙 =
𝒂𝒚 =
ሬሬሬԦ
𝚫𝒗𝒙
𝚫𝒕
𝚫𝐯𝒚
𝚫𝒕
= 𝒂 cos θ
= 𝒂 sin θ
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: INTRODUCTION TO PROJECTILE MOTION
● Projectile Motion occurs when an object is launched & moves in 2D under the influence of only ___________.
- Remember! Whenever we have Physics problems in 2D, we decompose them into 1D (X & Y).
1D
VERTICAL
MOTION
2D PROJECTILE MOTION
HORIZONTAL LAUNCH
DOWNWARD LAUNCH
UPWARD LAUNCH
● Projectile Motion COMBINES (1) horizontal motion where 𝑎𝑥 = ___, and (2) vertical motion where 𝑎𝑦 = ____.
EQUATIONS TO USE FOR PROJECTILE MOTION
UAM EQUATIONS
X (ax = 0)
Y (ay = –g)
(1) 𝐯𝐱 = 𝐯𝟎𝐱 + 𝐚𝐱 𝐭
(1) 𝐯𝐲 = 𝐯𝟎𝐲 + 𝐚𝐲 𝐭
(2) 𝐯𝐱𝟐 = 𝐯𝟎𝐱𝟐 + 𝟐𝐚𝐱 𝚫𝐱
(2) 𝐯𝐲𝟐 = 𝐯𝟎𝐲𝟐 + 𝟐𝐚𝐲 𝚫𝐲
𝟏
(3) 𝚫𝐱 = 𝐯𝟎𝐱 𝐭 + 𝐚𝐱 𝐭 𝟐
𝟐
𝟏
*(4) 𝚫𝐱 = (𝐯𝟎𝐱 + 𝐯𝐱 )𝐭
𝟐
𝟏
(3) 𝚫𝐲 = 𝐯𝟎𝐲 𝐭 + 𝐚𝐲 𝐭 𝟐
𝟐
𝟏
*(4) 𝚫𝐲 = (𝐯𝟎𝐲 + 𝐯𝐲 )𝐭
𝟐
VECTOR EQs
𝐴Ԧ
𝜃𝑥
𝐴𝑥
𝐴𝑦
𝑨 = √𝑨𝒙𝟐 + 𝑨𝒚𝟐
|𝑨𝒚 |
𝜽𝒙 = 𝐭𝐚𝐧−𝟏 (
)
|𝑨𝒙 |
𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝒙 )
𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝒙 )
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: Which of the following quantities are constant during projectile motion?
A)
B)
C)
D)
Vertical acceleration & vertical velocity
Angle (direction) of the velocity vector
Horizontal acceleration & vertical velocity
Vertical acceleration & horizontal velocity
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: SOLVING PROJECTILE MOTION PROBLEMS (WITH HORIZONTAL LAUNCH EXAMPLE)
EXAMPLE: A ball rolls horizontally off a 2m-tall table with a speed of 3.0 m/s. Calculate a) the time it takes for the ball to hit
the ground, and b) the horizontal displacement (range) of the ball
PROJECTILE MOTION
1) Draw paths in X&Y and points of interest
(Points of Interest: initial, final, max height, etc.)
2) Determine target variable
3) Determine interval and UAM equation
4) Solve
VECTOR EQs
UAM EQUATIONS
X
Y
𝐴Ԧ
(1) 𝐯𝐲 = 𝐯𝟎𝐲 + 𝐚𝐲 𝐭
(2) 𝐯𝐲𝟐 = 𝐯𝟎𝐲𝟐 + 𝟐𝐚𝐲 𝚫𝐲
𝚫𝐱 = 𝐯𝐱 𝐭
𝐴𝑦
𝜃𝑥
𝐴𝑥
𝟏
(3) 𝚫𝐲 = 𝐯𝟎𝐲 𝐭 + 𝐚𝐲 𝐭 𝟐
𝟐
𝑨 = √𝑨𝒙𝟐 + 𝑨𝒚𝟐
𝟏
*(4) 𝚫𝐲 = (𝐯𝟎𝐲 + 𝐯𝐟 )𝐭
𝟐
|𝑨𝒚 |
𝜽𝒙 = 𝐭𝐚𝐧−𝟏 (
)
|𝑨𝒙 |
𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝒙 )
𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝒙 )
● In projectile motion, time t can be found from either X OR Y axis equations. Always try the ___ axis equation first.
- If you get stuck and can't solve using X axis equation, always try to solve it with a Y axis equation, and vice versa.
● When an object is launched horizontally, its initial velocity is ONLY in the ___ axis: 𝒗𝟎𝒙 = ____ = ____
𝒗𝟎𝒚 = ____
- Remember: all objects in projectile motion always have (1) 𝒂𝒙 = 0, so 𝒗𝒙 ________ changes; and (2) 𝒂𝒚 = –g
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: A rock is thrown horizontally with a speed of 20 m/s from the edge of a high cliff. It lands 80 m from the cliff's
base. How tall is the cliff?
A)
B)
C)
D)
78.4 m
19.6 m
122.5 m
24.5 m
PROJECTILE MOTION
1) Draw paths in X&Y and points of interest
(Points of Interest: initial, final, max height, etc.)
2) Determine target variable
3) Determine interval and UAM equation
4) Solve
VECTOR EQs
UAM EQUATIONS
X
Y
𝐴Ԧ
(1) 𝐯𝐲 = 𝐯𝟎𝐲 + 𝐚𝐲 𝐭
(2) 𝐯𝐲𝟐 = 𝐯𝟎𝐲𝟐 + 𝟐𝐚𝐲 𝚫𝐲
𝚫𝐱 = 𝐯𝐱 𝐭
𝜃𝑥
𝐴𝑥
𝐴𝑦
𝟏
(3) 𝚫𝐲 = 𝐯𝟎𝐲 𝐭 + 𝐚𝐲 𝐭 𝟐
𝟐
𝑨 = √𝑨𝒙𝟐 + 𝑨𝒚𝟐
𝟏
*(4) 𝚫𝐲 = (𝐯𝟎𝐲 + 𝐯𝐟 )𝐭
𝟐
|𝑨𝒚 |
𝜽𝒙 = 𝐭𝐚𝐧−𝟏 (
)
|𝑨𝒙 |
𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝒙 )
𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝒙 )
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: A ping-pong player standing 1.6 m from the net serves the ball horizontally. The ball is hit 1.2 m above the floor.
What initial speed does the ball need to go over the net, which is 1.6m away from the player and 0.90m above the floor?
A)
B)
C)
D)
2.1 m/s
3.2 m/s
9.2 m/s
6.4 m/s
PROJECTILE MOTION
1) Draw paths in X&Y and points of interest
(Points of Interest: initial, final, max height, etc.)
2) Determine target variable
3) Determine interval and UAM equation
4) Solve
VECTOR EQs
UAM EQUATIONS
X
Y
𝐴Ԧ
(1) 𝐯𝐲 = 𝐯𝟎𝐲 + 𝐚𝐲 𝐭
(2)
𝚫𝐱 = 𝐯𝐱 𝐭
𝐯𝐲𝟐
=
𝐯𝟎𝐲𝟐
+ 𝟐𝐚𝐲 𝚫𝐲
𝜃𝑥
𝐴𝑥
𝐴𝑦
𝟏
(3) 𝚫𝐲 = 𝐯𝟎𝐲 𝐭 + 𝐚𝐲 𝐭 𝟐
𝟐
𝑨 = √𝑨𝒙𝟐 + 𝑨𝒚𝟐
𝟏
*(4) 𝚫𝐲 = (𝐯𝟎𝐲 + 𝐯𝐟 )𝐭
𝟐
|𝑨𝒚 |
𝜽𝒙 = 𝐭𝐚𝐧−𝟏 (
)
|𝑨𝒙 |
𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝒙 )
𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝒙 )
Page 16
Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: You kick a ball horizontally at 8 m/s from the roof of a 40m-tall building. Unfortunately, a car below you on the
street accelerates forwards from rest, and your ball lands on the car. What was the acceleration of the car?
A)
B)
C)
D)
7.92 m/s2
5.60 m/s2
1.96 m/s2
11.2 m/s2
PROJECTILE MOTION
1) Draw paths in X&Y and points of interest
(Points of Interest: initial, final, max height, etc.)
2) Determine target variable
3) Determine interval and UAM equation
4) Solve
VECTOR EQs
UAM EQUATIONS
X
Y
𝐴Ԧ
(1) 𝐯𝐲 = 𝐯𝟎𝐲 + 𝐚𝐲 𝐭
(2) 𝐯𝐲𝟐 = 𝐯𝟎𝐲𝟐 + 𝟐𝐚𝐲 𝚫𝐲
𝚫𝐱 = 𝐯𝐱 𝐭
𝜃𝑥
𝐴𝑥
𝐴𝑦
𝟏
(3) 𝚫𝐲 = 𝐯𝟎𝐲 𝐭 + 𝐚𝐲 𝐭 𝟐
𝟐
𝑨 = √𝑨𝒙𝟐 + 𝑨𝒚𝟐
𝟏
*(4) 𝚫𝐲 = (𝐯𝟎𝐲 + 𝐯𝐟 )𝐭
𝟐
|𝑨𝒚 |
𝜽𝒙 = 𝐭𝐚𝐧−𝟏 (
)
|𝑨𝒙 |
𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝒙 )
𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝒙 )
Page 17
Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: INTRO TO RELATIVE MOTION (RELATIVE VELOCITY)
● We always measure velocity relative to some ___________________, called a Frame of Reference.
- Frame of Reference is always the ground or the Earth unless otherwise stated.
EXAMPLE: You are an Observer measuring the velocities of A, B, & C (using a speed gun) on a moving walkway (like the
ones in airports) moving at 3m/s. If A stands still, B walks RIGHT at 2m/s, C walks LEFT at 2m/s, what does your speed
gun measure their velocities to be?
At t=0
At t=1
A is moving relative to you (Observer) at ____ m/s
A
B
O
(you)
–2m/s
+2m/s
B is moving relative to you (Observer) at ____ m/s
C
C is moving relative to you (Observer) at ____ m/s
𝒗𝑾 = +3m/s
Moving Walkway
● Whenever a stationary (not moving) object is inside/within/on top of a moving object, they have the _______ velocity.
- In other words, their velocities relative to all other reference frames is the same.
- Also, their velocities relative to each other is ______.
● Relative Velocity is really just the ______________________ of velocities.
Page 18
Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
CONCEPT: SOLVING RELATIVE VELOCITY PROBLEMS IN 1D
● To solve Relative Velocity problems, just add/subtract velocities in different reference frames:
Relative Velocity Equation
Velocity of A
Velocity of A
=
relative to C
relative to B
+
Velocity of B
relative to C
______=____________
→ Always set up so:
1) Inner subscripts of terms on right side of equation are the ________
2) Outer subscripts of terms on right side = subscripts of term on left side
EXAMPLE: You’re in a car moving at 45m/s east, relative to the ground. A truck is ahead of you, also moving east at 60m/s.
What is the velocity of the truck relative to your car?
RELATIVE VELOCITY IN 1D
1) Draw diagram, identify all objects & references
2) Write each given velocities w/ subscript notation
3) Write relative velocity equation according to rules
4) Solve
● If you’re given a velocity with the correct subscripts but in the opposite order, you can _________ the subscripts
- When you flip the subscripts, the sign of the number flips (positive ↔ negative): 𝒗𝑨𝑩 = −𝒗𝑩𝑨
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Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: A boat on a river is traveling from a pier to a point 500 m upstream (against the river’s current). The current
flows at 4 m/s. If the boat makes the trip in 250 s, what is the speed of the boat relative to the water?
A)
B)
C)
D)
2 m/s
0 m/s
6 m/s
12 m/s
RELATIVE VELOCITY IN 1D
1) Draw diagram, identify all objects & references
2) Write each given velocities w/ subscript notation
3) Write relative velocity equation according to rules
4) Solve
Page 20
Cutnell & Johnson - 12th edition - Physics
Ch. 03 - Kinematics in Two Dimensions
PROBLEM: City B lies directly east of city A. Without any wind, an airliner makes the 2775-km flight between them in 3.30h.
If a steady 225-km/h wind blows from west to east and the airliner has the same speed relative to the air as before, how
long will the trip from A to B take?
A)
B)
C)
D)
7.1 hrs
2.6 hrs
9.0 hrs
5.2 hrs
RELATIVE VELOCITY IN 1D
1) Draw diagram, identify all objects & references
2) Write each given velocities w/ subscript notation
3) Write relative velocity equation according to rules
4) Solve
Page 21