kinematics varying accelerations (1d) vectors...

KinematicsVarying Accelerations (1D)

Vectors (2D)

Lana Sheridan

De Anza College

Sept 29, 2017

Last time

• kinematic equations

• using kinematic equations

Overview

• falling objects and g

• varying acceleration

• vectors

• components

• vector addition and subtraction

Falling Objects

One common scenario of interest where acceleration is constant isfalling objects.

Objects (with mass) near the Earth’s surface accelerate at aconstant rate of g = 9.8 ms−2. (Or, about 10 ms−2) Thekinematics equations for constant acceleration all apply.

ExampleYou drop a rock off of a 50 m tall building. With what velocity doesit strike the ground? How long does it take to fall to the ground?

What quantities do we know? Which ones do we want to predict?

Which equation(s) should we use?

Falling Objects

ExampleYou drop a rock off of a 50 m tall building. With what velocity doesit strike the ground? How long does it take to fall to the ground?

What quantities do we know? Which ones do we want to predict?

Which equation(s) should we use?

Falling Objects

Figure 2.40 Vertical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearlywith time and that acceleration is constant. Misconception Alert! Notice that the position vs. time graph shows vertical position only. It is easy to get the impression thatthe graph shows some horizontal motion—the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is time, not space. Theactual path of the rock in space is straight up, and straight down.

Discussion

The interpretation of these results is important. At 1.00 s the rock is above its starting point and heading upward, since and are both

positive. At 2.00 s, the rock is still above its starting point, but the negative velocity means it is moving downward. At 3.00 s, both and

are negative, meaning the rock is below its starting point and continuing to move downward. Notice that when the rock is at its highest point (at

1.5 s), its velocity is zero, but its acceleration is still . Its acceleration is for the whole trip—while it is moving up andwhile it is moving down. Note that the values for are the positions (or displacements) of the rock, not the total distances traveled. Finally, note

that free-fall applies to upward motion as well as downward. Both have the same acceleration—the acceleration due to gravity, which remainsconstant the entire time. Astronauts training in the famous Vomit Comet, for example, experience free-fall while arcing up as well as down, as wewill discuss in more detail later.

Making Connections: Take-Home Experiment—Reaction Time

A simple experiment can be done to determine your reaction time. Have a friend hold a ruler between your thumb and index finger, separated byabout 1 cm. Note the mark on the ruler that is right between your fingers. Have your friend drop the ruler unexpectedly, and try to catch itbetween your two fingers. Note the new reading on the ruler. Assuming acceleration is that due to gravity, calculate your reaction time. How farwould you travel in a car (moving at 30 m/s) if the time it took your foot to go from the gas pedal to the brake was twice this reaction time?

CHAPTER 2 | KINEMATICS 65

1OpenStax Physics

Acceleration in terms of g

Because g is a constant, and because we have a good intuition forit, we can use it as a “unit” of acceleration.

(“How many times g is this acceleration?”)

Consider the drag race car in the earlier example. For the car themaximum acceleration was a = 26.0 m s−2. How many gs is that?

A 0

B roughly 1

C roughly 2 and a half

D 26

Acceleration of a Falling Object

Question

A baseball is thrown straight up. It reaches a peak height of 15 m,measured from the ground, in a time 1.7 s. Treating “up” as thepositive direction, what is the acceleration of the ball when itreached its peak height?

A 0 m/s2

B −8.8 m/s2

C +8.8 m/s2

D −9.8 m/s2

1Princeton Review: Cracking the AP Physics Exam

1-D Kinematics with varying acceleration

If we have a varying acceleration a(t), we should use calculus notthe kinematics equations.

In the homework (question 57), there is a situation where the jerk,J = da

dt is constant, but not zero.

In that case the acceleration is not constant:

∆a =

∫ t0J dt ′

so,a(t) = ai + Jt

The velocity cannot be found by using v(t) = vi + at !

Instead, integrate again.


If we have a varying acceleration a(t), we should use calculus notthe kinematics equations.

In the homework (question 57), there is a situation where the jerk,J = da

dt is constant, but not zero.

In that case the acceleration is not constant:

∆a =

∫ t0J dt ′

so,a(t) = ai + Jt

The velocity cannot be found by using (((((((hhhhhhhv(t) = vi + at !

Instead, integrate again.


Suppose you have a particle with a varying acceleration, but youdon’t know a as a function of t.

Instead, you know a as a function of position, x : a(x)

Can you use that knowledge to find changes in velocity,displacement, etc?

Yes!

Consider the integral: ∫a(x) dx

It’s not immediately obvious what it’s physical meaning is, but let’stry to investigate...


∫a dx =

∫dv

dtdx

Using the chain rule: ∫a dx =

∫dv

dx

dx

dtdx

Rearranging: ∫a dx =

∫dx

dt

(dv

dxdx

)Chain rule again, and using v = dx

dt :∫a dx =

∫v dv


∫ xfxi

a dx =

∫ vfvi

v dv =v2

2

∣∣∣∣vfvi

So, we find that∫a(x) dx does mean something physically:

1

2(v2f − v2i ) =

∫a dx

IF a is a constant:

1

2(v2f − v2i ) = a∆x ⇒ v2f = v2i + 2a∆x .

If a varies with x , all we need to do is evaluate the integral∫a(x) dx.


An asteroid falls in a straight line toward the Sun, starting fromrest when it is 1.00 million km from the Sun. Its acceleration isgiven by, a = − k

x2where x is the distance from the Sun to the

asteroid, and k = 1.33× 1020 m3/s2 is a constant.

After it has fallen through half-a-million km, what is its speed?

Math you will need for 2-Dimensions

Before going into motion in 2 dimensions, we will review somethings about vectors.

Vectors

scalar

A scalar quantity indicates an amount. It is represented by a realnumber. (Assuming it is a physical quantity.)

vector

A vector quantity indicates both an amount and a direction. It isrepresented more than one real number. (Assuming it is a physicalquantity.)

There are many ways to represent a vector.

• a magnitude and (an) angle(s)

• magnitudes in several perpendicular directions

Representing Vectors: AnglesBearing anglesExample, a plane flies 750 km h−1

at a bearing of 70◦

Generic reference anglesA baseball is thrown at 10ms−1 30◦ above the horizontal.

N

E

Representing Vectors: Unit Vectors

Magnitudes in several perpendicular directions: using unit vectors.

Unit vectors have a magnitude of one unit.

A set of perpendicular unit vectors defines a basis or decompositionof a vector space.

In two dimensions, a pair of perpendicular unit vectors are usuallydenoted i and j (or sometimes x, y).

Representing Vectors: Unit Vectors

Perpendicular unit vectors are usually denoted i and j (orsometimes x, y).

A generic 2 dimensional vector can be written as v = ai+ bj,where a and b are numbers.

Components

Consider the 2 dimensional vector A = Ax i+ Ay j, where Ax andAy are numbers.

We then say that Ax is the i-component (or x-component) of Aand Ay is the j-component (or y -component) of A.

3.4 Components of a Vector and Unit Vectors 65

3.4 Components of a Vector and Unit VectorsThe graphical method of adding vectors is not recommended whenever high accuracy is required or in three-dimensional problems. In this section, we describe a method of adding vectors that makes use of the projections of vectors along coordinate axes. These projections are called the components of the vec-tor or its rectangular components. Any vector can be completely described by its components. Consider a vector A

S lying in the xy plane and making an arbitrary angle u

with the positive x axis as shown in Figure 3.12a. This vector can be expressed as the sum of two other component vectors A

Sx , which is parallel to the x axis, and A

Sy , which

is parallel to the y axis. From Figure 3.12b, we see that the three vectors form a right triangle and that A

S5 A

Sx 1 A

Sy. We shall often refer to the “components

of a vector AS

,” written Ax and Ay (without the boldface notation). The compo-nent Ax represents the projection of A

S along the x axis, and the component Ay

represents the projection of AS

along the y axis. These components can be positive or negative. The component Ax is positive if the component vector A

Sx points in

the positive x direction and is negative if AS

x points in the negative x direction. A similar statement is made for the component Ay.

Use the law of sines (Appendix B.4) to find the direction of R

S measured from the northerly direction:

sin bB

5sin u

R

sin b 5BR

sin u 535.0 km48.2 km

sin 1208 5 0.629

b 5 38.9°

The resultant displacement of the car is 48.2 km in a direction 38.9° west of north.

Finalize Does the angle b that we calculated agree with an estimate made by looking at Figure 3.11a or with an actual angle measured from the diagram using the graphical method? Is it reasonable that the magnitude of R

S is larger

than that of both AS

and BS

? Are the units of RS

correct? Although the head to tail method of adding vectors works well, it suffers from two disadvantages. First, some

people find using the laws of cosines and sines to be awk-ward. Second, a triangle only results if you are adding two vectors. If you are adding three or more vectors, the resulting geometric shape is usually not a triangle. In Sec-tion 3.4, we explore a new method of adding vectors that will address both of these disadvantages.

Suppose the trip were taken with the two vectors in reverse order: 35.0 km at 60.0° west of north first and then 20.0 km due north. How would the magnitude and the direction of the resultant vector change?

Answer They would not change. The commutative law for vector addition tells us that the order of vectors in an addition is irrelevant. Graphically, Figure 3.11b shows that the vectors added in the reverse order give us the same resultant vector.

WHAT IF ?

Figure 3.12 (a) A vector AS

lying in the xy plane can be rep-resented by its component vectors AS

x and AS

y. (b) The y component vector A

Sy can be moved to the

right so that it adds to AS

x. The vector sum of the component vectors is A

S. These three vectors

form a right triangle.

y

xO

y

x

y

xO

x

y

u u

AS

AS

AS

AS

AS

AS

a b

▸ 3.2 c o n t i n u e d

Notice that Ax = A cos θ and Ay = A sin θ.

Components vs Magnitude / Angle Notation

Notice that Ax = A cos θ and Ay = A sin θ.


3.4 Components of a Vector and Unit VectorsThe graphical method of adding vectors is not recommended whenever high accuracy is required or in three-dimensional problems. In this section, we describe a method of adding vectors that makes use of the projections of vectors along coordinate axes. These projections are called the components of the vec-tor or its rectangular components. Any vector can be completely described by its components. Consider a vector A

S lying in the xy plane and making an arbitrary angle u

with the positive x axis as shown in Figure 3.12a. This vector can be expressed as the sum of two other component vectors A

Sx , which is parallel to the x axis, and A

Sy , which

is parallel to the y axis. From Figure 3.12b, we see that the three vectors form a right triangle and that A

S5 A

Sx 1 A

Sy. We shall often refer to the “components

of a vector AS

,” written Ax and Ay (without the boldface notation). The compo-nent Ax represents the projection of A

S along the x axis, and the component Ay

represents the projection of AS

along the y axis. These components can be positive or negative. The component Ax is positive if the component vector A

Sx points in

the positive x direction and is negative if AS

x points in the negative x direction. A similar statement is made for the component Ay.

Use the law of sines (Appendix B.4) to find the direction of R

S measured from the northerly direction:

sin bB

5sin u

R

sin b 5BR

sin u 535.0 km48.2 km

sin 1208 5 0.629

b 5 38.9°

The resultant displacement of the car is 48.2 km in a direction 38.9° west of north.

Finalize Does the angle b that we calculated agree with an estimate made by looking at Figure 3.11a or with an actual angle measured from the diagram using the graphical method? Is it reasonable that the magnitude of R

S is larger

than that of both AS

and BS

? Are the units of RS

correct? Although the head to tail method of adding vectors works well, it suffers from two disadvantages. First, some

people find using the laws of cosines and sines to be awk-ward. Second, a triangle only results if you are adding two vectors. If you are adding three or more vectors, the resulting geometric shape is usually not a triangle. In Sec-tion 3.4, we explore a new method of adding vectors that will address both of these disadvantages.

Suppose the trip were taken with the two vectors in reverse order: 35.0 km at 60.0° west of north first and then 20.0 km due north. How would the magnitude and the direction of the resultant vector change?

Answer They would not change. The commutative law for vector addition tells us that the order of vectors in an addition is irrelevant. Graphically, Figure 3.11b shows that the vectors added in the reverse order give us the same resultant vector.

WHAT IF ?

Figure 3.12 (a) A vector AS

lying in the xy plane can be rep-resented by its component vectors AS

x and AS

y. (b) The y component vector A

Sy can be moved to the

right so that it adds to AS

x. The vector sum of the component vectors is A

S. These three vectors

form a right triangle.

y

xO

y

x

y

xO

x

y

u u

AS

AS

AS

AS

AS

AS

a b

▸ 3.2 c o n t i n u e d

Also notice,A = |A| =

√Ax + Ay

and

θ = tan−1

(Ay

Ax

)if the angle is given as shown.

Why Vectors?

Of course, there is no reason to limit this to two dimensions.

With three dimensions we introduce another unit vector k = z.And we can have as many dimensions as we need by adding moreperpendicular unit vectors.

Vectors are the right tool for working in higher dimensions.

They have a property that correctly reflects what it means forthere to be more than one dimension: that each perpendiculardirection is independent of the others.

This makes life much easier: we will be able to solve for motion inthe x direction separately from motion in the y direction.

Visualizing Motion in 2 Dimensions

Imagine an air hockey puck moving with horizontally constantvelocity:

4.2 Two-Dimensional Motion with Constant Acceleration 81

dimensional) motion. Second, the direction of the velocity vector may change with time even if its magnitude (speed) remains constant as in two-dimensional motion along a curved path. Finally, both the magnitude and the direction of the velocity vector may change simultaneously.

Q uick Quiz 4.1 Consider the following controls in an automobile in motion: gas pedal, brake, steering wheel. What are the controls in this list that cause an acceleration of the car? (a) all three controls (b) the gas pedal and the brake (c) only the brake (d) only the gas pedal (e) only the steering wheel

4.2 Two-Dimensional Motion with Constant Acceleration

In Section 2.5, we investigated one-dimensional motion of a particle under con-stant acceleration and developed the particle under constant acceleration model. Let us now consider two-dimensional motion during which the acceleration of a particle remains constant in both magnitude and direction. As we shall see, this approach is useful for analyzing some common types of motion. Before embarking on this investigation, we need to emphasize an important point regarding two-dimensional motion. Imagine an air hockey puck moving in a straight line along a perfectly level, friction-free surface of an air hockey table. Figure 4.4a shows a motion diagram from an overhead point of view of this puck. Recall that in Section 2.4 we related the acceleration of an object to a force on the object. Because there are no forces on the puck in the horizontal plane, it moves with constant velocity in the x direction. Now suppose you blow a puff of air on the puck as it passes your position, with the force from your puff of air exactly in the y direction. Because the force from this puff of air has no component in the x direction, it causes no acceleration in the x direction. It only causes a momentary acceleration in the y direction, causing the puck to have a constant y component of velocity once the force from the puff of air is removed. After your puff of air on the puck, its velocity component in the x direction is unchanged as shown in Figure 4.4b. The generalization of this simple experiment is that motion in two dimen-sions can be modeled as two independent motions in each of the two perpendicular directions associated with the x and y axes. That is, any influence in the y direc-tion does not affect the motion in the x direction and vice versa. The position vector for a particle moving in the xy plane can be written

rS 5 x i 1 y j (4.6)

where x, y, and rS change with time as the particle moves while the unit vectors i and j remain constant. If the position vector is known, the velocity of the particle can be obtained from Equations 4.3 and 4.6, which give

vS 5d rS

dt5

dxdt

i 1dydt

j 5 vx i 1 vy j (4.7)

The horizontal red vectors, representing the x component of the velocity, are the same length in both parts of the figure, which demonstrates that motion in two dimensions can be modeled as two independent motions in perpendicular directions.

x

y

x

y

a

b

Figure 4.4 (a) A puck moves across a horizontal air hockey table at constant velocity in the x direction. (b) After a puff of air in the y direction is applied to the puck, the puck has gained a y com-ponent of velocity, but the x com-ponent is unaffected by the force in the perpendicular direction.

If it experiences a momentary upward (in the diagram)acceleration, it will have a component of velocity upwards.The horizontal motion remains unchanged!

Vectors Properties and Operations

EqualityVectors A = B if and only if the magnitudes and directions are thesame. (Each component is the same.)

AdditionA+ B

62 Chapter 3 Vectors

Q uick Quiz 3.1 Which of the following are vector quantities and which are scalar quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass

3.3 Some Properties of VectorsIn this section, we shall investigate general properties of vectors representing physi-cal quantities. We also discuss how to add and subtract vectors using both algebraic and geometric methods.

Equality of Two VectorsFor many purposes, two vectors A

S and B

S may be defined to be equal if they have

the same magnitude and if they point in the same direction. That is, AS

5 BS

only if A ! B and if A

S and B

S point in the same direction along parallel lines. For exam-

ple, all the vectors in Figure 3.5 are equal even though they have different starting points. This property allows us to move a vector to a position parallel to itself in a diagram without affecting the vector.

Adding VectorsThe rules for adding vectors are conveniently described by a graphical method. To add vector B

S to vector A

S, first draw vector A

S on graph paper, with its magni-

tude represented by a convenient length scale, and then draw vector BS

to the same scale, with its tail starting from the tip of A

S, as shown in Figure 3.6. The resultant

vector RS

5 AS

1 BS

is the vector drawn from the tail of AS

to the tip of BS

. A geometric construction can also be used to add more than two vectors as shown in Figure 3.7 for the case of four vectors. The resultant vector R

S 5 A

S 1 B

S 1

CS

1 DS

is the vector that completes the polygon. In other words, RS

is the vector drawn from the tail of the first vector to the tip of the last vector. This technique for adding vectors is often called the “head to tail method.” When two vectors are added, the sum is independent of the order of the addi-tion. (This fact may seem trivial, but as you will see in Chapter 11, the order is important when vectors are multiplied. Procedures for multiplying vectors are dis-cussed in Chapters 7 and 11.) This property, which can be seen from the geometric construction in Figure 3.8, is known as the commutative law of addition:

AS

1 BS

5 BS

1 AS

(3.5)Commutative law of addition X

O

y

x

Figure 3.5 These four vectors are equal because they have equal lengths and point in the same direction.

AS

BS

CS

DS

AS BS

CS DS

RS!

""

"Figure 3.7 Geometric construc-tion for summing four vectors. The resultant vector R

S is by definition

the one that completes the polygon.

AS

BS

BS

AS AS

BS

BS

RS !

!"

"

Draw , then add .

AS

BS

AS

Draw , then add .A

S

BS

Figure 3.8 This construction shows that A

S1 B

S5 B

S1 A

S or, in

other words, that vector addition is commutative.

Pitfall Prevention 3.1Vector Addition Versus Scalar Addition Notice that AS

1 BS

5 CS

is very different from A " B ! C. The first equa-tion is a vector sum, which must be handled carefully, such as with the graphical method. The second equation is a simple alge-braic addition of numbers that is handled with the normal rules of arithmetic.

Figure 3.6 When vector BS

is added to vector A

S, the resultant R

S is

the vector that runs from the tail of AS

to the tip of BS

.

!"

AS

RS A

S BS

BS

Vectors Properties and OperationsProperties of Addition

• A+ B = B+ A (commutative)

• (A+ B) + C = A+ (B+ C) (associative) 3.3 Some Properties of Vectors 63

When three or more vectors are added, their sum is independent of the way in which the individual vectors are grouped together. A geometric proof of this rule for three vectors is given in Figure 3.9. This property is called the associative law of addition:

AS

1 1 BS

1 CS 2 5 1 A

S1 B

S 2 1 CS

(3.6)

In summary, a vector quantity has both magnitude and direction and also obeys the laws of vector addition as described in Figures 3.6 to 3.9. When two or more vectors are added together, they must all have the same units and they must all be the same type of quantity. It would be meaningless to add a velocity vector (for example, 60 km/h to the east) to a displacement vector (for example, 200 km to the north) because these vectors represent different physical quantities. The same rule also applies to scalars. For example, it would be meaningless to add time intervals to temperatures.

Negative of a VectorThe negative of the vector A

S is defined as the vector that when added to A

S gives

zero for the vector sum. That is, AS

1 12 AS 2 5 0. The vectors A

S and 2 A

S have the

same magnitude but point in opposite directions.

Subtracting VectorsThe operation of vector subtraction makes use of the definition of the negative of a vector. We define the operation A

S2 B

S as vector 2 B

S added to vector A

S:

AS

2 BS

5 AS

1 12 BS 2 (3.7)

The geometric construction for subtracting two vectors in this way is illustrated in Figure 3.10a. Another way of looking at vector subtraction is to notice that the difference AS

2 BS

between two vectors AS

and BS

is what you have to add to the second vector

WW Associative law of addition

Add and ; then add theresult to .A

S

AS

AS

AS

BS

BS

BS

BS

BS

!

CS

CS

CS

Add and ; then add to the result.

AS

BS

CS

!

CS

AS BS !

!

CS )

(

AS BS !

!

CS

)

(

Figure 3.9 Geometric construc-tions for verifying the associative law of addition.

"

AS

AS

AS

BS

BS

BS

BS

"

AS

BS

CS

# "

AS

BS

We would draw here if we were adding it to .

Adding " to is equivalent to subtracting from .

BS

AS

AS

BS

a b

AS

BS

Vector is the vector we must add to to obtain .

AS

BS

CS

# "

Figure 3.10 (a) Subtracting vector B

S from vector A

S. The vec-

tor 2 BS

is equal in magnitude to vector B

S and points in the oppo-

site direction. (b) A second way of looking at vector subtraction.

62 Chapter 3 Vectors

Q uick Quiz 3.1 Which of the following are vector quantities and which are scalar quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass

3.3 Some Properties of VectorsIn this section, we shall investigate general properties of vectors representing physi-cal quantities. We also discuss how to add and subtract vectors using both algebraic and geometric methods.

Equality of Two VectorsFor many purposes, two vectors A

S and B

S may be defined to be equal if they have

the same magnitude and if they point in the same direction. That is, AS

5 BS

only if A ! B and if A

S and B

S point in the same direction along parallel lines. For exam-

ple, all the vectors in Figure 3.5 are equal even though they have different starting points. This property allows us to move a vector to a position parallel to itself in a diagram without affecting the vector.

Adding VectorsThe rules for adding vectors are conveniently described by a graphical method. To add vector B

S to vector A

S, first draw vector A

S on graph paper, with its magni-

tude represented by a convenient length scale, and then draw vector BS

to the same scale, with its tail starting from the tip of A

S, as shown in Figure 3.6. The resultant

vector RS

5 AS

1 BS

is the vector drawn from the tail of AS

to the tip of BS

. A geometric construction can also be used to add more than two vectors as shown in Figure 3.7 for the case of four vectors. The resultant vector R

S 5 A

S 1 B

S 1

CS

1 DS

is the vector that completes the polygon. In other words, RS

is the vector drawn from the tail of the first vector to the tip of the last vector. This technique for adding vectors is often called the “head to tail method.” When two vectors are added, the sum is independent of the order of the addi-tion. (This fact may seem trivial, but as you will see in Chapter 11, the order is important when vectors are multiplied. Procedures for multiplying vectors are dis-cussed in Chapters 7 and 11.) This property, which can be seen from the geometric construction in Figure 3.8, is known as the commutative law of addition:

AS

1 BS

5 BS

1 AS

(3.5)Commutative law of addition X

O

y

x

Figure 3.5 These four vectors are equal because they have equal lengths and point in the same direction.

AS

BS

CS

DS

AS BS

CS DS

RS!

""

"

Figure 3.7 Geometric construc-tion for summing four vectors. The resultant vector R

S is by definition

the one that completes the polygon.

AS

BS

BS

AS AS

BS

BS

RS !

!"

"

Draw , then add .

AS

BS

AS

Draw , then add .A

S

BS

Figure 3.8 This construction shows that A

S1 B

S5 B

S1 A

S or, in

other words, that vector addition is commutative.

Pitfall Prevention 3.1Vector Addition Versus Scalar Addition Notice that AS

1 BS

5 CS

is very different from A " B ! C. The first equa-tion is a vector sum, which must be handled carefully, such as with the graphical method. The second equation is a simple alge-braic addition of numbers that is handled with the normal rules of arithmetic.

Figure 3.6 When vector BS

is added to vector A

S, the resultant R

S is

the vector that runs from the tail of AS

to the tip of BS

.

!"

AS

RS A

S BS

BS


Doing addition:Almost always the right answer is to break each vector intocomponents and sum each component independently.


which lies on the x axis and has magnitude 0Ax 0 . Likewise, AS

y 5 Ay jS

is the com-ponent vector of magnitude 0Ay 0 lying on the y axis. Therefore, the unit-vector notation for the vector A

S is

AS

5 Ax i 1 Ay j (3.12)

For example, consider a point lying in the xy plane and having Cartesian coordi-nates (x, y) as in Figure 3.15. The point can be specified by the position vector rS, which in unit-vector form is given by

rS 5 x i 1 y j (3.13)

This notation tells us that the components of rS are the coordinates x and y. Now let us see how to use components to add vectors when the graphical method is not sufficiently accurate. Suppose we wish to add vector B

S to vector A

S in Equa-

tion 3.12, where vector BS

has components Bx and By. Because of the bookkeeping convenience of the unit vectors, all we do is add the x and y components separately. The resultant vector R

S5 A

S1 B

S is

RS

5 1Ax i 1 Ay j 2 1 1Bx i 1 By j 2or

RS

5 1Ax 1 Bx 2 i 1 1Ay 1 By 2 j (3.14)

Because RS

5 Rx i 1 Ry j, we see that the components of the resultant vector are

Rx 5 Ax 1 Bx

Ry 5 Ay 1 By (3.15)

Therefore, we see that in the component method of adding vectors, we add all the x components together to find the x component of the resultant vector and use the same process for the y components. We can check this addition by components with a geometric construction as shown in Figure 3.16. The magnitude of R

S and the angle it makes with the x axis are obtained from its

components using the relationships

R 5 "Rx2 1 Ry

2 5 "1Ax 1 Bx 22 1 1Ay 1 By 22 (3.16)

tan u 5Ry

Rx5

Ay 1 By

Ax 1 Bx (3.17)

At times, we need to consider situations involving motion in three component directions. The extension of our methods to three-dimensional vectors is straight-forward. If A

S and B

S both have x, y, and z components, they can be expressed in

the form

AS

5 Ax i 1 Ay j 1 Az k (3.18)

BS

5 Bx i 1 By j 1 Bz k (3.19)

The sum of AS

and BS

is

RS

5 1Ax 1 Bx 2 i 1 1Ay 1 By 2 j 1 1Az 1 Bz 2 k (3.20)

Notice that Equation 3.20 differs from Equation 3.14: in Equation 3.20, the resul-tant vector also has a z component Rz ! Az " Bz. If a vector R

S has x, y, and z com-

ponents, the magnitude of the vector is R 5 !Rx2 1 Ry

2 1 Rz2. The angle ux

that RS

makes with the x axis is found from the expression cos ux ! Rx/R, with simi-lar expressions for the angles with respect to the y and z axes. The extension of our method to adding more than two vectors is also straight-forward. For example, A

S1 B

S1 C

S5 1Ax 1 Bx 1 Cx 2 i 1 1Ay 1 By 1 Cy 2 j 11Az 1 Bz 1 Cz 2 k. We have described adding displacement vectors in this section

because these types of vectors are easy to visualize. We can also add other types of

y

xO

(x, y)

y

x

j

i

rS

Figure 3.15 The point whose Cartesian coordinates are (x, y) can be represented by the position vector rS 5 x i 1 y j.

y

x

Bx

Ay

Ax

Rx

ByRy

AS

BS

RS

Figure 3.16 This geometric construction for the sum of two vectors shows the relationship between the components of the resultant R

S and the components

of the individual vectors.

Pitfall Prevention 3.3Tangents on Calculators Equa-tion 3.17 involves the calculation of an angle by means of a tangent function. Generally, the inverse tangent function on calculators provides an angle between #90° and "90°. As a consequence, if the vector you are studying lies in the second or third quadrant, the angle measured from the positive x axis will be the angle your calcu-lator returns plus 180°.

Vectors Properties and OperationsDoing addition:Almost always the right answer is to break each vector intocomponents and sum each component independently.

Examplew = 5 m at 36.9◦ above the horizontal.u = 17 m at 28.1◦ above the horizontal.

w + u = ?


NegationIf u = −v then u has the same magnitude as v but points in the

opposite direction.

SubtractionA− B = A+ (−B)

3.3 Some Properties of Vectors 63

When three or more vectors are added, their sum is independent of the way in which the individual vectors are grouped together. A geometric proof of this rule for three vectors is given in Figure 3.9. This property is called the associative law of addition:

AS

1 1 BS

1 CS 2 5 1 A

S1 B

S 2 1 CS

(3.6)

In summary, a vector quantity has both magnitude and direction and also obeys the laws of vector addition as described in Figures 3.6 to 3.9. When two or more vectors are added together, they must all have the same units and they must all be the same type of quantity. It would be meaningless to add a velocity vector (for example, 60 km/h to the east) to a displacement vector (for example, 200 km to the north) because these vectors represent different physical quantities. The same rule also applies to scalars. For example, it would be meaningless to add time intervals to temperatures.

Negative of a VectorThe negative of the vector A

S is defined as the vector that when added to A

S gives

zero for the vector sum. That is, AS

1 12 AS 2 5 0. The vectors A

S and 2 A

S have the

same magnitude but point in opposite directions.

Subtracting VectorsThe operation of vector subtraction makes use of the definition of the negative of a vector. We define the operation A

S2 B

S as vector 2 B

S added to vector A

S:

AS

2 BS

5 AS

1 12 BS 2 (3.7)

The geometric construction for subtracting two vectors in this way is illustrated in Figure 3.10a. Another way of looking at vector subtraction is to notice that the difference AS

2 BS

between two vectors AS

and BS

is what you have to add to the second vector

WW Associative law of addition

Add and ; then add theresult to .A

S

AS

AS

AS

BS

BS

BS

BS

BS

!

CS

CS

CS

Add and ; then add to the result.

AS

BS

CS

!

CS

AS BS !

!

CS )

(

AS BS !

!

CS

)

(

Figure 3.9 Geometric construc-tions for verifying the associative law of addition.

"

AS

AS

AS

BS

BS

BS

BS

"

AS

BS

CS

# "

AS

BS

We would draw here if we were adding it to .

Adding " to is equivalent to subtracting from .

BS

AS

AS

BS

a b

AS

BS

Vector is the vector we must add to to obtain .

AS

BS

CS

# "

Figure 3.10 (a) Subtracting vector B

S from vector A

S. The vec-

tor 2 BS

is equal in magnitude to vector B

S and points in the oppo-

site direction. (b) A second way of looking at vector subtraction.

Summary

• free fall

• varying acceleration

• vectors vs scalars

• representing vectors

• some vector operations: addition, subtraction

Collected Homework! due Friday, Oct 6.

(Uncollected) Homework Serway & Jewett,

• NEW: Ch 2, onward from page 49. Probs: 53, 57, 59

• Ch 3, onward from page 71. Objective Qs 1 & 3; Probs: 31,55, 65

kinematics varying accelerations (1d) vectors...

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