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