Chapter 3 Vectors and Two-Dimensional Motion. Vectors Vectors – physical quantities having both magnitude and direction Vectors are labeled either a or.

Chapter 3

Vectors and Two-Dimensional Motion

Vectors

• Vectors – physical quantities having both magnitude and direction

• Vectors are labeled either a or

• Vector magnitude is labeled either |a| or a

• Two (or more) vectors having the same magnitude and direction are identical

a

Vector sum (resultant vector)

• Not the same as algebraic sum

• Triangle method of finding the resultant:a) Draw the vectors “head-to-tail”b) The resultant is drawn from the tail of A to the head of B

A

B

R = A + B

Addition of more than two vectors

• When you have many vectors, just keep repeating the process until all are included

• The resultant is still drawn from the tail of the first vector to the head of the last vector

Commutative law of vector addition

A + B = B + A

Associative law of vector addition

(A + B) + C = A + (B + C)

Negative vectors

Vector (- b) has the same magnitude as b but opposite direction

Vector subtraction

Special case of vector addition: A - B = A + (- B)

Multiplying a vector by a scalar

• The result of the multiplication is a vector

c A = B

• Vector magnitude of the product is multiplied by the scalar

|c| |A| = |B|

• If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector

Vector components

• Component of a vector is the projection of the vector on an axis

• To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector

Vector components

x

y

yx A

AAAA tan

22

inAAAA yx s cos

Unit vectors

• Unit vector:A) Has a magnitude of 1 (unity)B) Lacks both dimension and unitC) Specifies a direction

• Unit vectors in a right-handed coordinate system

Adding vectors by components

In 2D case:

jbibb

jaiaa

yx

yx

ˆˆ

ˆˆ

bar

yyy

xxx

bar

bar

Chapter 3Problem 14

A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

Position

The position of an object is described by its position vector,

kzjyixr ˆˆˆ

r

Displacement

The displacement of the object is defined as the change in its position,

if rrr

)ˆˆ()ˆˆ( jyixjyixr iiff

jyyixx ififˆ)(ˆ)(

jyixr ˆˆ

r

Velocity

• Average velocity

• Instantaneous velocity

t

jyix

t

rvavg

ˆˆ

t

rv

t

0lim

jvivv yxˆˆ

Instantaneous velocity

Vector of instantaneous velocity is always tangential to the object’s path at the object’s position

Acceleration

• Average acceleration

• Instantaneous acceleration

t

v

t

vva if

avg

t

va

t

0lim

Acceleration

• Acceleration – the rate of change of velocity (vector)

• The magnitude of the velocity (the speed) can change – tangential acceleration

• The direction of the velocity can change – radial acceleration

• Both the magnitude and the direction can change

Projectile motion

• A special case of 2D motion

• An object moves in the presence of Earth’s gravity

• We neglect the air friction and the rotation of the Earth

• As a result, the object moves in a vertical plane and follows a parabolic path

• The x and y directions of motion are treated independently

Projectile motion – X direction

• A uniform motion: ax = 0

• Initial velocity is

• Displacement in the x direction is described as

iixi vv cos

tvxx iii )cos(

Projectile motion – Y direction

• Motion with a constant acceleration: ay = – g

• Initial velocity is

• Therefore

• Displacement in the y direction is described as

iiyi vv sin

2

2

1)sin( gttvyy iii

gtvv iiy sin

Projectile motion: putting X and Y together

constvv iix cos

2

2

1)sin( gttvyy iii

gtvv iy 0sin

tvxx iii )cos(

Projectile motion: trajectory and range

ii

g

vR 2sin

2

2

2

)cos(2)(tan

iii v

gxxy

ii

g

vh 2

2

sin2

Projectile motion: trajectory and range

ii

g

vR 2sin

2

2

2

)cos(2)(tan

iii v

gxxy

ii

g

vh 2

2

sin2

Chapter 3Problem 58

A 2.00-m-tall basketball player is standing on the floor 10.0 m from the basket. If he shoots the ball at a 40.0° angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket is 3.05 m.

Relative motion

• Reference frame: physical object and a coordinate system attached to it

• Reference frames can move relative to each other

• We can measure displacements, velocities, accelerations, etc. separately in different reference frames

Relative motion

• If reference frames A and B move relative to each other with a constant velocity

• Then

• Acceleration measured in both reference frames will be the same

BAPBPA vvv

BAPBPA rrr

PBPA aa

BAv

Questions?

Answers to the even-numbered problems

Chapter 3

Problem 2: (a) Approximately 484 km (b) Approximately 18.1° N of W

Answers to the even-numbered problems

Chapter 3

Problem 6: (a) Approximately 6.1 units at 113°(b) Approximately 15 units at 23°

Answers to the even-numbered problems

Chapter 3

Problem 10: 1.31 km north, 2.81 km east

Answers to the even-numbered problems

Chapter 3

Problem 28: x = 7.23 × 103 m, y = 1.68 × 103 m