Chapter 3 Vectors & 2-Dimensional Motion 1

Chapter 3Vectors & 2-Dimensional

Motion

Chapter 3 Vectors & 2-Dimensional Motion 2

3.1 Vectors & Scalars Revisited Vector: magnitude & direction

Displacement Velocity Acceleration

Scalar: magnitude but no direction Temperature Speed Time intervals

Chapter 3 Vectors & 2-Dimensional Motion 3

3.2 Vector Properties Vector Format

Handwritten: A Printed: A, bold font

Scalar Format: A, italics

Chapter 3 Vectors & 2-Dimensional Motion 4

3.2 Vector Properties Vector Equality

A & B are equal if they have the same magnitude & direction.

Equal vectors can be moved parallel to itself without being affected

Chapter 3 Vectors & 2-Dimensional Motion 5

3.2 Vector Properties Which of these vectors have the same

MAGNITUDE?

Chapter 3 Vectors & 2-Dimensional Motion 6

3.2 Vector Properties Which of these vectors have the same

DIRECTION?

Chapter 3 Vectors & 2-Dimensional Motion 7

3.2 Vector Properties Adding Vectors

Must have same units Graphical Methods

Triangular method of addition Parallelogram method of addition Sum is independent of order of addition

A + B = B + A Commutative law of addition

Component Method

Chapter 3 Vectors & 2-Dimensional Motion 8

3.2 Vector Properties Triangle Method

Chapter 3 Vectors & 2-Dimensional Motion 9

3.2 Vector Properties Parallelogram Method

Chapter 3 Vectors & 2-Dimensional Motion 10

3.2 Vector Properties Negative of a Vector

Same magnitude opposite direction A + (-A) = 0

Subtracting Vectors A – B = A + (-B)

Multiplying/Dividing by a scalar 4A, A/5

Chapter 3 Vectors & 2-Dimensional Motion 11

3.2 Vector Properties• Adding 2 Vectors • Adding 3 Vectors• Subtracting Vectors

Chapter 3 Vectors & 2-Dimensional Motion 12

3.3 Vector Components V = Vx + Vy

Vx = V cos Ө Vy = V sin Ө

Vx

Vy

Chapter 3 Vectors & 2-Dimensional Motion 13

3.3 Vector Components

http://id.mind.net/~zona/mstm/physics/mechanics/vectors/components/vectorComponents.html

Chapter 3 Vectors & 2-Dimensional Motion 14

Vector Tutorial

Khan Academy Vector Tutorial

Aircraft Takeoff Problem

Chapter 3 Vectors & 2-Dimensional Motion 15

Practice Problems Find the x and y components of the following

vectors:240 N at 330º34 m/s at 210º15 m at 12º

20 m/s2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion 16

Practice Problems Find the x and y components of the following

vectors:240 N at 330º Fy = 207.85 Fx = 120

34 m/s at 210º15 m at 12º

20 m/s2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion 17

Practice Problems Find the x and y components of the following

vectors:240 N at 330º Fy = 207.85 Fx = 120

34 m/s at 210º Vy = 17.0 Vx = 29.44

15 m at 12º 20 m/s2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion 18

Practice Problems Find the x and y components of the following

vectors:240 N at 330º Fy = 207.85 Fx = 120

34 m/s at 210º Vy = 17.0 Vx = 29.44

15 m at 12º xy = 3.12 xx = 1.4 20 m/s2 at 90º

Chapter 3 Vectors & 2-Dimensional Motion 19

Practice Problems Find the x and y components of the following

vectors:240 N at 330º Fy = -120 Fx =

207.85 34 m/s at 210º Vy = 17.0 Vx = 29.44

15 m at 12º xy = 3.12 xx = 1.4 20 m/s2 at 90º ay = 20.0 ax = 0

Component Method• Adding vectors using “trig” & “arithmetic”Step 1: Find all x and y components Step 2: Add up all the x components

Add up all the y componentsStep 3: Using the “new” x and y components

find the “new” resulting vector!

Step 4: Sanity check

21

Vector & Projectile Motion Practice Problems Find the resultant of the following 2 vectors:

1) 100 units due west and 2) 200 units 30o north of east.

22

Vector & Projectile Motion Practice Problems Find the resultant of the following 2 vectors:

1) 100 units due east and 2) 200 units 30o north of east.

124 units 54o north of west

23

Vector & Projectile Motion Practice Problems An ant on a picnic table travels 30 cm

eastward, then 25 cm northward and finally 15 cm westward. What is its directional displacement with respect to its original position?

24

Vector & Projectile Motion Practice Problems An ant on a picnic table travels 30 cm

eastward, then 25 cm northward and finally 15 cm westward. What is its directional displacement with respect to its original position?

59o north of east

25

Vector & Projectile Motion Practice Problems A boy pulls a sled across a level field by

exerting a force of 110 newtons at an angle of 30o with the ground. What are the parallel and perpendicular components, respectively, of this force with respect to the ground?

26

Vector & Projectile Motion Practice Problems A boy pulls a sled across a level field by

exerting a force of 110 newtons at an angle of 30o with the ground. What are the parallel and perpendicular components, respectively, of this force with respect to the ground?

95 newtons, 55 newtons

27

Vector & Projectile Motion Practice Problems I walk 6 miles in a straight line in a direction

north of east and I end up 2 miles east and several miles north. How many degrees north of east have I walked?

28

Vector & Projectile Motion Practice Problems I walk 6 miles in a straight line in a direction

north of east and I end up 2 miles east and several miles north. How many degrees north of east have I walked?

71o

Chapter 3 Vectors & 2-Dimensional Motion 29

Practice Problems From the x and y components given, find

the direction and magnitude of the resultant.Fy = 120 N, Fx = 345 Nvy = 31 m/s, vx = 8 m/s

Chapter 3 Vectors & 2-Dimensional Motion 30

Practice Problems A soccer ball is kicked with a horizontal

velocity of 11.3 m/s and a vertical velocity of 3.5 m/s. What is the magnitude and direction of the ball's velocity?

A shot putter applies a force of 415 N to a shot at an angle of 37º. What are the horizontal and vertical components of this force?

Homework

Page(s) 76 & 77#1,2,5,7,10,13,15,18,19

Due Tomorrow whether you have class or not!

Chapter 3 Vectors & 2-Dimensional Motion 32

Projectile Motion

Chapter 3 Vectors & 2-Dimensional Motion 33

Chapter 3Projectile Motion

Chapter 3 Vectors & 2-Dimensional Motion 34

Chapter 3Projectile Motion Animated Projectile Motion

Chapter 3 Vectors & 2-Dimensional Motion 35

3.5 Projectile Motion Can be described as a superposition of two

independent motions in the x and y directions If air resistance is negligible, horizontal component

remains constant because there is no acceleration in the horizontal direction.

Vertical component is equal to the free-fall acceleration, g.

Vertical component of velocity and y-direction displacement are identical to a freely falling object.

Chapter 3 Vectors & 2-Dimensional Motion 36

3.5 Projectile Motion If you are carrying a ball and running at

constant speed and wish to throw the ball so that you can catch it as it comes back down, should you (a) throw the ball at a 45o angle above the horizontal and maintain the same speed, (b) throw the ball straight up in the air and slow down to catch it, or (c) throw the ball straight up in the air and maintain the same speed?

Chapter 3 Vectors & 2-Dimensional Motion 37

3.5 Projectile Motion If you are carrying a ball and running at

constant speed and wish to throw the ball so that you can catch it as it comes back down, should you (a) throw the ball at a 45o angle above the horizontal and maintain the same speed, (b) throw the ball straight up in the air and slow down to catch it, or (c) throw the ball straight up in the air and maintain the same speed?

Chapter 3 Vectors & 2-Dimensional Motion 38

3.5 Projectile Motion As a projectile moves in its parabolic path,

the velocity and acceleration vectors are perpendicular to each other (a) everywhere along its path, (b) at the peak of its path, (c) nowhere along its path, or (d) not enough information is given.

Chapter 3 Vectors & 2-Dimensional Motion 39

3.5 Projectile Motion As a projectile moves in its parabolic path,

the velocity and acceleration vectors are perpendicular to each other (a) everywhere along its path, (b) at the peak of its path, (c) nowhere along its path, or (d) not enough information is given.

Chapter 3 Vectors & 2-Dimensional Motion 40

3.5 Projectile Motion A home run is hit into the stands. The ball is

hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Chapter 3 Vectors & 2-Dimensional Motion 41

3.5 Projectile Motion A home run is hit into the stands. The ball is

hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Chapter 3 Vectors & 2-Dimensional Motion 42

3.5 Projectile Motion A home run is hit into the stands. The ball is

hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Chapter 3 Vectors & 2-Dimensional Motion 43

3.5 Projectile Motion A home run is hit into the stands. The ball is

hit from home plate into the center field stands along a parabolic path. What is the acceleration of the ball (a) while it is rising, (b) at the highest point of the trajectory, and (c) while it is descending after reaching the highest point? Ignore air resistance.

Could It Happen?

In the movie Speed a bus traveling at nearly 68 mph is rigged with a bomb that will go off if the bus goes below 50 mph. It has to jump a 50’ gap in a bridge – could it be done?

Simple explanation

More involved Physics explanation

45

Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is thrown

from shoulder height at an initial velocity of 29.4 m/s at an initial angle of 30o with respect to the horizontal. It is in the air for a total time interval of 3 s before it is caught by the 3rd baseman at shoulder height level. What is the ball’s horizontal displacement?

46

Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is thrown

from shoulder height at an initial velocity of 29.4 m/s at an initial angle of 30o with respect to the horizontal. It is in the air for a total time interval of 3 s before it is caught by the 3rd baseman at shoulder height level. What is the ball’s horizontal displacement?

76.4 m

47

Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is

released from shoulder height at an initial velocity of 29.4 m/s at initial angle of 30o with respect to the horizontal. What is the maximum vertical displacement that the ball reaches during its trajectory?

48

Vector & Projectile Motion Practice Problems A baseball thrown from the outfield is

released from shoulder height at an initial velocity of 29.4 m/s at initial angle of 30o with respect to the horizontal. What is the maximum vertical displacement that the ball reaches during its trajectory?

11.0 m

49

Vector & Projectile Motion Practice Problems A stone is thrown at an angle of 30o above

the horizontal from the top edge of a cliff with an initial speed of 12 m/s. A stop watch measures the stone’s trajectory time from the top of the cliff to the bottom to be 5.6 s. What is the height of the cliff?

50

Vector & Projectile Motion Practice Problems A stone is thrown at an angle of 30o above

the horizontal from the top edge of a cliff with an initial speed of 12 m/s. A stop watch measures the stone’s trajectory time from the top of the cliff to the bottom to be 5.6 s. What is the height of the cliff?

120 m

51

Vector & Projectile Motion Practice Problems A bridge that was 5 m long has been washed

out by the rain several days ago. How fast must a car be going to successfully jump the stream? Although the road is level on both sides of the bridge, the road on the far side is 2 m lower than the road on this side.

52

Vector & Projectile Motion Practice Problems A bridge that was 5 m long has been washed

out by the rain several days ago. How fast must a car be going to successfully jump the stream? Although the road is level on both sides of the bridge, the road on the far side is 2 m lower than the road on this side.

8 m/s

53

Vector & Projectile Motion Practice Problems A track star in the broad jump goes into the

jump at 12 m/s and launches herself at 20o above the horizontal. How long is she in the air before returning to the ground?

54

Vector & Projectile Motion Practice Problems A track star in the broad jump goes into the

jump at 12 m/s and launches herself at 20o above the horizontal. How long is she in the air before returning to the ground?

0.83 s

55

Vector & Projectile Motion Practice Problems A fireman, 50 m away from a burning

building, directs a stream of water from a hose at an angle of 30o above the horizontal. If the velocity of the stream of water is 40 m/s, at what height will the stream of water strike the building?

56

Vector & Projectile Motion Practice Problems A fireman, 50 m away from a burning

building, directs a stream of water from a hose at an angle of 30o above the horizontal. If the velocity of the stream of water is 40 m/s, at what height will the stream of water strike the building?

18.7 m