motion in two dimensions and vectors (chapter 3) 3 (motion in... · 2019-08-27 · horizontal. what...

Everglades High SchoolPhysics Honors

Motion in Two Dimensions and Vectors

(Chapter 3)

Alberto Dominguez

Updated for 2018-9 Edition of HMH Physics

Topics

• Vectors

• Scalars and vectors

• Properties of Vectors

• Vector Operations

• Coordinate Systems

• Determine Resultant Magnitude and Direction

• Resolving Vectors into Components

• Adding Vectors that are not Perpendicular

Topics

• Projectile Motion

•Two Dimensional Motion

• Relative Motion

•Frames of Reference

•Relative Velocity

Section 1 Objectives

• SC.912.P12.1 – Distinguish between scalar and

vector quantities and assess which should be

used to describe an event

• SC.912.P.12.2 – Analyze the motion of an object

in terms of its position, velocity and acceleration

• MAFS.912.N-VM.1.3 – Solve problems involving

velocity and other quantities that can be

represented as vectors

• Add and subtract vectors using the graphical

method.

• Multiply and divide vectors by scalars.

Section 1 Introduction to Vectors

p. 82

Scalars and Vectors

• Vectors indicate direction; scalars do not

• Scalar = physical quantity that has magnitude but

no direction (e.g., distance, speed)

• Vector = physical quantity that has both magnitude

and direction (e.g., displacement, velocity)

• Vectors are represented by boldface v or an

arrow over the variable 𝑣

• Resultant = the sum of two or more vectors

• Vectors can be added graphically


p. 82

Properties of Vectors

• Vectors can be moved parallel to

themselves in a diagram

• Vectors can be added in any order

• To subtract a vector, add its

negative/opposite

• Multiplying or dividing a vector by a

scalar results in a vector


p. 84

Multiplying a Vector by a Scalar


Reference: www.matlabassignments.com

Section 1 Review #2 – Solve by Graphical Method

A roller coaster moves 85 m

horizontally, then travels 45 m at

an angle of 30.0º above the

horizontal. What is its

displacement from its starting

point?


p. 85

Section 1 Review #3

A novice pilot sets a plane’s

controls thinking the plane will fly

250 km/h to the north. If the wind

blows at 75 km/h toward the

southeast, what is the plane’s

resultant velocity?


p. 85


• SC.912.P.12.2 – Analyze the motion of an object in

terms of its position, velocity and acceleration

• MAFS.912.N-VM.1.3 – Solve problems involving

velocity and other quantities that can be represented as

vectors

• Identify appropriate coordinate systems for solving

problems with vectors.

• Apply the Pythagorean Theorem and tangent function

to calculate the magnitude and direction of a resultant

vector.

• Resolve vectors into components using the sine and

cosine functions.

• Add vectors that are not perpendicular.

Section 2 Vector Operations

p. 86

Coordinate Systems in Two Dimensions


Determining Resultant Magnitude and Direction


Using the Pythagorean Theorem

p. 86, Figure 2.4


Using the Tangent Function

p. 88, Figure 2.5


Resolving Vectors into Components

• Components = the x and y pieces of a vector

• Sine of an angle = opposite leg / hypotenuse

• sin 𝜃 =𝑜𝑝𝑝

ℎ𝑦𝑝

• sin 𝜃 =𝑦

𝑟

• 𝑦 = 𝑟 sin 𝜃


p. 90

Resolving Vectors into Components

• Cosine of an angle = adjacent leg / hypotenuse

• cos 𝜃 =adj

ℎ𝑦𝑝

• cos 𝜃 =x

𝑟

• x = 𝑟 cos 𝜃


p. 90

Practice Problem A2

While following the directions on a

treasure map, a pirate walks 45.0

m north and then walks 7.5 m

east. What single straight-line

displacement could the pirate

have taken to reach the treasure?


p. 89

Practice Problem A3

Emily passes a soccer ball 6.0 m

directly across the field to Kara.

Kara then kicks the ball 14.5 m

directly down the field to Luisa.

What is the ball’s total

displacement as it travels

between Emily and Luisa?


p. 89

Practice Problem A4

A hummingbird flying above the

ground flies 1.2 m along a straight

path. Upon spotting a flower below,

the hummingbird drops directly

downward 1.4 m to hover in front of

the flower. What is the

hummingbird’s total displacement?


p. 89

#1

Practice A – Finding Resultant Magnitude and Direction

Chapter Review

#21, #22, #23

p. 89


Practice Problem B3

A truck drives up a hill with

a 15º incline. If the truck

has a constant speed of 22

m/s, what are the horizontal

and vertical components of

the truck’s velocity?


p. 92

#1, #2, #4

Practice B – Resolving Vectors

Chapter Review

#24, #25

p. 92


Adding Vectors that are Not Perpendicular – Algebraic Method

Textbook Page 93 – Sample Problem C

1) Select a coordinate system. Then sketch

and label each vector.

2) Find the x and y components of all vectors.

3) Add the x and y components.

4) Use the Pythagorean Theorem to find the

magnitude of the resultant vector.

5) Use a suitable trigonometric function to find

the angle.


p. 93

Practice Problem C1

A football player runs directly down

the field for 35 m before turning to

the right at an angle of 25º from his

original direction and running an

additional 15 m before getting

tackled. What is the magnitude and

direction of the runner’s total

displacement?


p. 94

Practice Problem C2

A plane travels 2.5 km at an

angle of 35º to the ground and

then travels at 5.2 km at an angle

of 22º to the ground. What is the

magnitude and direction of the

plane’s total displacement?


p. 94

Practice Problem C3

During a rodeo, a clown runs 8.0 m

north, turns 55º north of east, and

runs 3.5 m. Then, after waiting for

the bull to come near, the clown

turns due east and runs 5.0 m to exit

the arena. What is the clown’s total

displacement?


p. 94

Practice Problem C4 (modified)

An airplane flying parallel to the

ground undergoes two

consecutive displacements. The

first is 75 km 25.0º west of north,

and the second is 155 km 60.0º

east of north. What is the total

displacement of the airplane?


p. 94

None

Practice C – Adding Vectors Algebraically

Chapter Review

#26

p. 94


Chapter Review #8

A dog searching for a bone

walks 3.50 m south, then 8.20

m at an angle of 30.0º north of

east, and finally 15.0 m west.

Find the dog’s resultant

displacement.


p. 111

Chapter Review #9

A man lost in a maze makes three

consecutive displacements so that at

the end of the walk he is back where

he started. The first displacement is

8.00 m westward, and the second is

13.0 m northward. Find the third

displacement.


p. 111

Law of Cosines

b a

A B

C

c

𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶


Law of Sines

b a

A B

C

c

sin 𝐴

𝑎=

sin𝐵

𝑏=

sin 𝐶

𝑐


Practice Problem

A plane travels at 450 mph with a

direction of 70° north of east. If there

is a 100 mph wind to the north, what

are the new direction and ground

speed of the plane if no corrections

are made for the wind?


Practice Problem

Two planes leave from the same

airport at the same time. One plane

has an air speed of 500 mph and a

direction of 80° S of E, and the

second has an air speed of 600 mph

and a direction of 35° N of E. After 2

hours, how far apart are the planes?



• SC.912.P.12.2 – Analyze the motion of an

object in terms of its position, velocity and

acceleration

• Recognize examples of projectile motion.

• Describe the path of a projectile as a

parabola.

• Resolve vectors into their components and

apply the kinematic equations to solve

problems involving projectile motion.

Section 3 Projectile Motion

p. 95

Two Dimensional Motion

• Use of components avoids vector multiplication

• There is a statement in your book that is false

• “Velocity, acceleration and displacement do not all

point in the same direction.”

• That’s correct, but now the book says

• “This makes the vector forms of the equations

difficult to solve.”

• That is absolutely incorrect; we will use the

vector forms of many equations because they

are EASIER to solve


p. 95


• Use of components avoids vector multiplication

• Recall x = vit + ½ at2 from chapter 2

• In vector notation, that becomes

» 𝑥 = 𝑣𝑖𝑡 +1

2 𝑎𝑡2

• In this case, it’s easier to work in components

»𝑥 = 𝑣𝑖𝑥𝑡 +1

2𝑎𝑥𝑡

2

»𝑦 = 𝑣𝑖𝑦𝑡 +1

2𝑎𝑦𝑡

2

»𝑧 = 𝑣𝑖𝑧𝑡 +1

2𝑎𝑧𝑡

2 (we′re only going to do x and y)


p. 95


• Components simplify projectile motion

• We can analyze x and y components separately

• Most problems follow a two-step pattern, either

»We solve an x equation and then use the result in

the y equation, or

»We solve a y equation and then use the result in

the x equation

• Projectile motion – study of the path taken by

objects thrown into the air and subject to gravity


p. 95


• Projectiles follow parabolic trajectories

• Objects with a forward speed do not fall

straight down like they do in cartoons

• If an object has an initial horizontal

velocity, there will be horizontal motion

throughout the flight of the projectile

• Projectile motion is free fall with an

initial horizontal velocity

• We will ignore air resistance


p. 96

Figure 16


p. 97, Figure 3.4

Let’s Review Chapter 2

A ball is dropped from a

height of 0.70 m above

the ground. How long

does it take to hit the

ground?


Practice Problem D1

A baseball rolls off a 0.70 m

high desk and strikes the

floor 0.25 m away from the

base of the desk. How fast

was the ball rolling?• Solve the y equation and then use the

result in the x equation


p. 99

Practice Problem D2 (modified)

Tom chases Jerry across a 1.0 m

high table. Jerry steps out of the

way, and Tom slides off the table

and strikes the floor 2.2 m from the

edge of the table. How long was

Tom’s flight through the air and what

was his speed when he slid off the

table?


p. 99

Example

Cambridge Physics Coursebook, p. 31, Based on Worked Example 9

Justin Bieber is thrown horizontally at 12.0 m/s

from the top of a cliff 40.0 m high.

a) Calculate how long Justin Bieber takes to

reach the ground.

2.86 s

b) Calculate how far Justin Bieber lands from

the base of the cliff.

34.3 m


#3, #4

Practice D – Projectiles Launched Horizontally

Chapter Review

#32, #33

p. 99



• Use components to analyze objects launched at an

angle.

• 𝑣𝑥𝑖 = 𝑣 cos 𝜃

• 𝑣𝑦𝑖 = 𝑣 sin 𝜃


p. 99

Problem

In a scene in an action movie, a

stuntman jumps from the top of one

building to the top of another building 4.0

m away. After a running start, he leaps

at a velocity of 5.0 m/s at an angle of 15º

with respect to the flat roof. Will he

make it to the other roof, which is 2.5 m

lower than the building he jumps from?


Practice Problem E3

A baseball is thrown at an angle

of 25º relative to the ground at a

speed of 23.0 m/s. If the ball

was caught 42.0 m from the

thrower, how long was it in the

air, and how high above the

thrower did the ball travel?


p. 101

Practice Problem E4

Salmon often jump waterfalls to

reach their breeding grounds.

One salmon starts 2.00 m from a

waterfall that is 0.55 m tall and

jumps at an angle of 32º. What

must be the salmon’s minimum

speed to reach the waterfall?


p. 101

Chapter Review #36 Part 1

When a water gun is fired while

being held horizontally at a

height of 1.00 m above ground

level, the water travels a

horizontal distance of 5.00 m.

What is the velocity of the water

when it leaves the gun?


p. 113

Chapter Review #36 Part 2

A child is holding the same water

gun horizontally and is sliding down

a 45º decline at a constant speed of

2.00 m/s. If the child fires from 1.00

m above the ground and the water

takes 0.39 s to reach the ground,

how far will it travel horizontally?


p. 113

Chapter Review #47

A ball player hits a home run, and the baseball just

clears the wall 21.0 m high located 130 m from home

plate. The ball is hit at an angle of 35º to the horizontal.

Air resistance is negligible. Assume the ball is hit at a

height of 1.00 m above the ground.

a) What is the initial speed of the ball?

b) How much time does it take for the ball to reach the

wall?

c) Find the components of the velocity of the ball when

it reaches the wall.

p. 114


Practice Problem

(Very difficult)

Jordan Spieth hits a golf ball at

an angle of 25º to the ground.

If the golf ball covers a

horizontal distance of 301.5 m,

what is the ball’s maximum

height?


#1, #2

Practice E – Projectiles Launched at an Angle

Chapter Review

#34, #35

p. 101


Section 3 Review #2

During a thunderstorm, a tornado lifts a

car to a height of 125 m above the

ground. Increasing in strength, the

tornado flings the car horizontally with a

speed of 90.0 m/s.

a) How long does the car take to reach

the ground?

b) How far horizontally does the car travel

before hitting the ground?


p. 101

Chapter Review #31 (re-written)

The fastest recorded pitch* in MLB

history was thrown by Aroldis Chapman

on 9/24/2010, who broke Nolan Ryan’s

previous record set in 1974. The pitch

was thrown horizontally, and the ball fell

0.744 m by the time it reached home

plate 18.3 m away. How fast was

Chapman’s pitch?

* Record book as of 9/17/2018


p. 112

Chapter Review #37

A ship maneuvers to within 2.50 x 103 m of

an island’s 1.80 x 103 m high mountain peak

and fires a projectile at an enemy ship 6.10

x 102 m on the other side of the peak. The

ship fires with an initial velocity of 2.50 x 102

m/s at an angle of 75º.

a) How close to the enemy ship does the

projectile land?

b) How close (vertically) does the projectile

come to the peak?


p. 113


• Describe situations in terms

of frame of reference.

• Solve problems involving

relative velocity.

Section 4 Relative Motion

p. 102

Frames of Reference and Relative Velocity

• Velocity measurements differ in different

frames of reference

• Remember the station and the train

• If a person is moving towards the front of the train,

vps = vts + vpt

• If a person is moving towards the back of the train,

vps = vts – vpt

• Aircraft Carrier – why does a plane launch

from the bow and land at the stern?


p. 103

Practice Problem

A boat heading north crosses a

wide river with a velocity of 10.0

km/hr relative to the water. The

river has a uniform velocity of

5.00 km/hr due east. Determine

the boat’s velocity with respect to

an observer on shore.


Practice Problem

A passenger at the rear of a train

traveling at 15 m/s relative to the

Earth throws a baseball with a speed

of 15 m/s in the direction opposite

the motion of the train. What is the

velocity of the baseball relative to the

Earth as it leaves the thrower’s

hand?


Practice Problem F2

A spy runs from the front to the

back of an aircraft carrier at a

velocity of 3.5 m/s. If the aircraft

carrier is moving forward at 18.0

m/s, how fast does the spy

appear to be running when

viewed by a stationary observer?


p. 105

#1, #3, #4

Practice F – Relative Velocity

Chapter Review

#43, #44, #45, #46

p. 105


Section 4 Review #1

A woman on a bicycle travels at 9

m/s relative to the ground as she

passes a boy on a tricycle going in

the opposite direction. If the boy is

traveling at 1 m/s relative to the

ground, how fast does the boy

appear to be moving relative to the

woman?


p. 105

Special Relativity and Velocity

Physics on the Edge Special Relativity

p. 106

Coming soon…

Theory of Special Relativity

Think Science Hypothesis or Theory?

p. 109

Coming soon…

Additional Practice

• Mixed Review #48 – #62

• Standardized Test Prep #1 – #17

motion in two dimensions and vectors (chapter 3) 3 (motion in... · 2019-08-27 · horizontal. what...

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