MP3ff

Assignment - Chapter 3

Due: 11:59pm on Thursday, January 28, 2010

Note: You will receive no credit for late submissions. To learn more, read your instructor's Grading Policy

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Tactics Box 3.1 Determining the Components of a Vector

Learning Goal: To practice Tactics Box 3.1 Determining the Components of a Vector.

When a vector is decomposed into component vectors and parallel to the coordinate axes, we can describe each component vector with a single number (a scalar) called the component. This tactics

box describes how to determine the x component and y component of vector , denoted and .

TACTICS BOX 3.1 Determining the components of a vector

The absolute value of the x component is the magnitude of the component vector .1.

The sign of is positive if points in the positive x direction; it is negative if points in the negative x direction.2.

The y component is determined similarly.3.

Part A

What is the magnitude of the component vector shown in the figure?

Express your answer in meters to one significant figure.

ANSWER: = 5

Correct

Part B

What is the sign of the y component of vector shown in the figure?

ANSWER: positive

negative

Correct

Part C

Now, combine the information given in the tactics box above to find the x and y components, and , of vector shown in the figure.

Express your answers, separated by a comma, in meters to one significant figure.

MasteringPhysics: Assignment Print View http://session.masteringphysics.com/myct/assignmentPrint?assignmentID...

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ANSWER: , = -2,-5

Correct ,

Vector Components--Review

Learning Goal: To introduce you to vectors and the use of sine and cosine for a triangle when resolving components.

Vectors are an important part of the language of science, mathematics, and engineering. They are used to discuss multivariable calculus, electrical circuits with oscillating currents, stress and strain in structures

and materials, and flows of atmospheres and fluids, and they have many other applications. Resolving a vector into components is a precursor to computing things with or about a vector quantity. Becauseposition, velocity, acceleration, force, momentum, and angular momentum are all vector quantities, resolving vectors into components is the most important skill required in a mechanics course.

The figure shows the components of , and , along the x and y axes of the coordinate system, respectively. The components of a vector

depend on the coordinate system's orientation, the key being the angle between the vector and the coordinate axes, often designated .

Part A

The figure shows the standard way of measuring the angle. is measured to the vector from the x axis, and counterclockwise is positive.

Express and in terms of the length of the vector and the angle , with the components separated by a comma.

ANSWER: , =

Correct

In principle, you can determine the components of any vector with these expressions. If lies in one of the other quadrants of the plane, will be an angle larger than 90 degrees (or in radians) and

and will have the appropriate signs and values.

Unfortunately this way of representing , though mathematically correct, leads to equations that must be simplified using trig identities such as

and

.

These must be used to reduce all trig functions present in your equations to either or . Unless you perform this followup step flawlessly, you will fail to recoginze that

,

and your equations will not simplify so that you can progress further toward a solution. Therefore, it is best to express all components in terms of either or , with between 0 and 90

degrees (or 0 and in radians), and determine the signs of the trig functions by knowing in which quadrant the vector lies.

Part B

When you resolve a vector into components, the components must have the form or . The signs depend on which quadrant the vector lies in, and there will be one component with

and the other with .

In real problems the optimal coordinate system is often rotated so that the x axis is not horizontal. Furthermore, most vectors will not lie in the first quadrant. To assign the sine and cosine correctly for vectorsat arbitrary angles, you must figure out which angle is and then properly reorient the definitional triangle.

As an example, consider the vector shown in the diagram labeled "tilted axes," where you know the angle between and the y axis.

Which of the various ways of orienting the definitional triangle must be used to resolve into components in the tilted coordinate system shown?

(In the figures, the hypotenuse is orange, the side adjacent to is red, and the side opposite is yellow.)


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Hint B.1 Recommended procedure for resolving a vector into components

Hint not displayed

Hint B.2 Finding the trigonometric functions

Hint not displayed

Indicate the number of the figure with the correct orientation.

ANSWER: 1

2

3

4

Correct

Part C

Choose the correct procedure for determining the components of a vector in a given coordinate system from this list:

ANSWER: Align the adjacent side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle.

Align the hypotenuse of a right triangle with the vector and an adjacent side along a coordinate direction with as the included angle.

Align the opposite side of a right triangle with the vector and the hypotenuse along a coordinate direction with as the included angle.

Align the hypotenuse of a right triangle with the vector and the opposite side along a coordinate direction with as the included angle.

Correct

Part D

The space around a coordinate system is conventionally divided into four numbered quadrants depending on the signs of the x and y coordinates .Consider the following conditions:

, A.

, B.

, C.

, D.

Which of these conditions are true in which quadrants?

Write the answer in the following way: If A were true in the third quadrant, B in the second, C in the first, and D in the fourth,

enter "3, 2, 1, 4" as your response.

ANSWER: 1,4,2,3

Correct

Part E

Now find the components and of in the tilted coordinate system of Part B.

Express your answer in terms of the length of the vector and the angle , with the components separated by a comma.


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ANSWER: , =

Correct

Adding and Subtracting Vectors Conceptual Question

Six vectors (A to F) have the magnitudes and directions indicated in the figure.

Part A

Which two vectors, when added, will have the largest x component?

Hint A.1 Largest x component

Hint not displayed

ANSWER: C and E

E and F

A and F

C and D

B and D

Correct

Part B

Which two vectors, when added, will have the largest y component?

Hint B.1 Largest y component

Hint not displayed

ANSWER: C and D

A and F

E and F

A and B

E and D

Correct

Part C

Which two vectors, when subtracted (i.e., when one vector is subtracted from the other), will have the largest magnitude?

Hint C.1 Subtracting vectors

Hint not displayed

ANSWER: A and F

A and E

D and B

C and D

E and F

Correct

Components of Vectors

Shown is a 10 by 10 grid, with coordinate axes x and y.

The grid runs from -5 to 5 on both axes. Drawn on this grid are four vectors, labeled through . This problem will ask you various questions

about these vectors. All answers should be in decimal notation, unless otherwise specified.


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Part A

What is the x component of ?

Hint A.1 How to derive the component

Hint not displayed

Express your answer to two significant figures.

ANSWER: = 2.5

Correct

Part B

What is the y component of ?

Express your answer to the nearest integer.

ANSWER: = 3

Correct

Part C

What is the y component of ?

Hint C.1 Consider the direction

Hint not displayed


ANSWER: = -3

Correct

Part D

What is the component of ?

Hint D.1 How to find the start and end points of the vector components

Hint not displayed


ANSWER: = -2

Correct

The following questions will ask you to give both components of vectors using the ordered pairs method. In this method, the x component is written first, followed by a comma, and then the y component. For

example, the components of would be written 2.5,3 in ordered pair notation.

The answers below are all integers, so estimate the components to the nearest whole number.

Part E

In ordered pair notation, write down the components of vector .

Express your answers to the nearest integer.

ANSWER: , = 2,-3

Correct

Part F

In ordered pair notation, write down the components of vector .

Express your answers to the nearest integer.

ANSWER: , = 2,-3

Correct

Part G

What is true about and ? Choose from the pulldown list below.

ANSWER:

Correct

Tracking a Plane


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A radar station, located at the origin of xz plane, as shown in the figure , detects an airplane coming straight at the station from the east. At first

observation (point A), the position of the airplane relative to the origin is . The position vector has a magnitude of 360 and is located at

exactly 40 above the horizon. The airplane is tracked for another 123 in the vertical east-west plane for 5.0 , until it has passed

directly over the station and reached point B. The position of point B relative to the origin is (the magnitude of is 880 ). The contact

points are shown in the diagram, where the x axis represents the ground and the positive z direction is upward.

Part A

Define the displacement of the airplane while the radar was tracking it: . What are the components of ?

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Finding the components of

Hint not displayed

Hint A.3 Finding the components of

Hint not displayed

Express in meters as an ordered pair, separating the x and z components with a comma, to two significant figures.

ANSWER: = -1100,26

Correct

Vector Addition

Consider the following three vectors:

,

,

and

.

Calculate the following combinations. Express your answers as ordered triplets [e.g., ].

Part A

Hint A.1 How to add vectors

Hint not displayed

ANSWER: = 5,-1,6

Correct

Part B

ANSWER: = 4,4,3

Correct

Part C

ANSWER: = 6,3,4

Correct

Part D

Hint D.1 Remember the order of precedence

Hint not displayed

ANSWER: = 8,5,-1

Correct

Part E


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ANSWER: = 14,2,15

Correct

Part F

ANSWER: = 16,10,11

Correct

Vector Addition and Subtraction

In general it is best to conceptualize vectors as arrows in space, and then to make calculations with them using their components. (You must first specify a coordinate system in order to find the components of

each arrow.) This problem gives you some practice with the components.

Let vectors , , and . Calculate the following, and express your answers as ordered triplets of values separated by commas.

Part A

ANSWER: = 3,-5,-4

Correct

Part B

ANSWER: = -5,4,0

Correct

Part C

ANSWER: = -6,4,3

Correct

Part D

ANSWER: = -3,-2,-11

Correct

Part E

ANSWER: = -11,14,8

Correct

Part F

ANSWER: = 17,-12,-6

Correct

Vector Addition in a Unit Cell

In nature, substances often possess a crystalline structure. The basic component of a crystal is the unit cell, such as the rectangular parallelpiped illustrated in the figure . (For the purposes of this problem, take

, , and to be positive.)

Find the following quantities. When the question asks for a vector, express your answer as an ordered triplet. That is, a vector with length in the x

direction, length in the y direction, and length in the z direction should be written as .

Part A

What is the vector from point O to point E?

ANSWER: =

Correct


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Part B

What is the vector from point O to point F?

ANSWER: =

Correct

Part C

What is the vector from point O to point C?

ANSWER: =

Correct

Part D

Hint D.1 How to use the components

Hint not displayed

ANSWER: =

Correct

Part E

Hint E.1 The meaning of | |

Hint not displayed

Hint E.2 Length of a vector

Hint not displayed

ANSWER: | | =

Correct

Part F

Hint F.1 Consider geometry

Hint not displayed

ANSWER: | | =

Correct

Part G

ANSWER: | | =

Correct

Vector Magnitude and Direction Conceptual Question

A man out walking his dog makes one complete pass around a perfectly square city block. He starts at point A and walks clockwise around the block.

Let be the displacement vector from A to B, be the displacement vector from B to C, etc.

Part A

Which of the following vectors is equal to ?


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Hint A.1 Determining a vector

Hint not displayed

Hint A.2 Equal vectors

Hint not displayed

ANSWER: only

only

only

None of the above

All of the above

Correct

Recall that, for vectors to be equal, they must have the same magnitude and direction.

Part B


ANSWER: only

only

only

None of the above

All of the above

Correct

Part C


Hint C.1 Determining the difference of two vectors

Hint not displayed

ANSWER: only

only

only

None of the above

All of the above

Correct

Vector Addition Ranking Task

Six vectors ( through ) have the magnitudes and directions indicated in the figure.

Part A

Rank the vector combinations on the basis of their magnitude.

Hint A.1 Adding vectors graphically

Hint not displayed

Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:


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View

Correct

Part B

Rank the vector combinations on the basis of their angle, measured counterclockwise from the positive x axis. Vectors parallel to the positive x axis have an angle of 0 . All angle measures fall between 0

and 360 .

Hint B.1 Angle of a vector

Hint not displayed

Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:

View

Correct

Adding Scalar Multiples of Vectors Graphically

Draw the vectors indicated. You may use any extra (unlabeled) vectors that are helpful; but, keep in mind that the unlabeled vectors should not be part of your submission.

Part A

Draw the vector .

Hint A.1 How to approach the problem

Hint not displayed

Hint A.2 Draw

Hint not displayed

Hint A.3 Adding vectors graphically

Hint not displayed

The length and orientation of the vector will be graded. The location of the vector is not important.

ANSWER:

View Correct

Now use the same technique to answer the next two parts.

Part B

Draw the vector .

Hint B.1 Find and

Hint not displayed

Hint B.2 Adding vectors graphically

Hint not displayed


ANSWER:


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View

Correct

Part C

Draw the vector .

Hint C.1 Find and

Hint not displayed

Hint C.2 Adding vectors graphically

Hint not displayed


ANSWER:

View

Correct

Biking Vectors

A student bikes to school by traveling first = 1.10 north, then = 0.600 west, and finally = 0.100 south.

Part A

If a bird were to start out from the origin (where the student starts) and fly directly (in a straight line) to the school, what distance would the bird cover?

Hint A.1 How to approach the problem by finding components

Hint not displayed

Hint A.2 Find the y component of

Hint not displayed

Hint A.3 Find the x component of

Hint not displayed

Hint A.4 Magnitude of a vector

Hint not displayed

Express your answer in miles.

ANSWER: = 1.17

Correct

This direct distance is sometimes called the distance "as the crow flies."

Part B

You will now find the same quantity algebraically, without the need to use much geometry. Take the north direction as the positive y direction and east as positive x. The origin is still where the student startsbiking.

Let be the displacement vector corresponding to the first leg of the student's trip. Express in component form.

Express your answer as two numbers separated by a comma (e.g., 1.0,2.0). By convention, the x component is written first.

ANSWER: , = 0,1.10

Correct

Part C

Similarly, let be the displacement vector corresponding to the second leg of the student's trip. Express in component form.

Express your answer as two numbers separated by a comma. Be careful with your signs.

ANSWER: , = -0.600,0

Correct

Part D

Finally, let be the displacement vector corresponding to the last leg of the student's trip. Express in component form.



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ANSWER: , = 0,-0.100

Correct

Part E

The displacement vector for the bird can be written as (see the figure ). In the space provided, type in component form.

Hint E.1 How to add vectors algebraically

Hint not displayed


ANSWER: , = -0.600,1.00

Correct

The magnitude of a vector with components and is given by

.

Using this definition, you can check that this approach yields the same value for as the one found earlier. Depending on the individual vectors in a given situation, you can decide whether the

geometric or the algebraic approach would be more suitable.

Part F

Finally, find , the angle north of west of the path followed by the bird.

Hint F.1 Method 1: Simple trigonometry

Hint not displayed

Hint F.2 Method 2: Vector equations

Hint not displayed

Express your answer numerically in degrees.

ANSWER: = 59.0

Correct

Resolving Vector Components with Trigonometry

Often a vector is specified by a magnitude and a direction; for example, a rope with tension exerts a force of magnitude in a direction 35 north

of east. This is a good way to think of vectors; however, to calculate results with vectors, it is best to select a coordinate system and manipulate the

components of the vectors in that coordinate system.

Part A

Find the components of the vector with length = 1.00 and angle =15.0 with respect to the x axis as shown.

Hint A.1 What is the x component?

Hint not displayed

Enter the x component followed by the y component, separated by a comma.

ANSWER: = 0.966,0.259

Correct

Part B


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Find the components of the vector with length = 1.00 and angle =15.0 with respect to the x axis as shown.

Hint B.1 What is the x component?

Hint not displayed


ANSWER: = 0.966,0.259

Correct

The components of still have the same form, that is, , despite 's placement with respect to the y axis on the drawing.

Part C

Find the components of the vector with length = 1.00 and angle 35.0 as shown.

Hint C.1 Method 1: Find the angle that makes with the positive x axis

Hint not displayed

Hint C.2 Method 2: Use vector addition

Hint not displayed


ANSWER: = -0.574,0.819

Correct

Problem 3.3

Part A

What is the x-component of vector in terms of the angle and the magnitude shown in the figure?

ANSWER: =

Correct

Part B

What is the y-component of vector in terms of the angle and the magnitude shown in the figure?

ANSWER: =

Correct

Part C

For the same vector, what is the x-component in terms of the angle and the magnitude ?

ANSWER: =

Correct

Part D

For the same vector, what is the y-component in terms of the angle and the magnitude ?

ANSWER: =

Correct

Problem 3.4

A position vector in the first quadrant has an x-component of and a magnitude of .

Part A


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What is the value of its y-component?

ANSWER: 8.00

Correct m

Problem 3.17

Are the following statements true or false?

Part A

The magnitude of a vector can be different in different coordinate systems.

ANSWER: True

False

Correct

Part B

The direction of a vector can be different in different coordinate systems.

ANSWER: True

False

Correct

Part C

The components of a vector can be different in different coordinate systems.

ANSWER: True

False

Correct

Problem 3.19

What are the x- and y-components of the velocity vector shown in the figure?

Part A

ANSWER: = -86.6

Correct m/s

Part B

ANSWER: =

50.0

Correct m/s

Problem 3.26

Part A

What is the angle between vectors and in the figure?


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ANSWER: = 71.6

Correct

Part B

Use components to determine the magnitude of .

ANSWER: = 3.00

Correct

Part C

Use components to determine the direction of .

Find , the angle between and the x-axis.

ANSWER: = 90.0

Correct

Problem 3.30

Carlos runs with velocity for minutes.

Part A

How far to the north of his starting position does Carlos end up?

ANSWER: 1270

Correct m

Problem 3.32

The minute hand on a watch is in length. What is the displacement vector of the tip of the minute hand

Part A

From 8:00 to 8:20 a.m.?

Express vector in the form , , where the x and y components are separated by a comma.

ANSWER: = 1.73,-3.00

Correct cm

Part B

From 8:00 to 9:00 a.m.?

Express vector in the form , , where the x and y components are separated by a comma.

ANSWER: = 0,0

Correct cm

Problem 3.37

Jack and Jill ran up the hill at . The horizontal component of Jill's velocity vector was .

Part A

What was the angle of the hill?

Give your answer to two significant figures.

ANSWER: = 33.6

Correct

Part B

What was the vertical component of Jill's velocity?

Give your answer to two significant figures.

ANSWER: =

1.66

Correct m/s

Problem 3.42

A flock of ducks is trying to migrate south for the winter, but they keep being blown off course by a wind blowing from the west at . A wise elder duck finally realizes that the solution is to fly at an

angle to the wind.


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Part A

If the ducks can fly at relative to the air, what direction should they head in order to move directly south?

ANSWER: 48.6

Correct west of south

Problem 3.43

The car in the figure speeds up as it turns a quarter-circle curve from north to east. When exactly halfway around the curve, the car's acceleration is

(2.0 , 15 south of east).

Part A

At this point, what is the component of tangent to the circle?

Express your answer using two significant figures.

ANSWER: =

1.0

Correct

Part B

At this point, what is the component of perpendicular to the circle?


ANSWER: = 1.7

Correct

Problem 3.44

The figure shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second ropewith 5.0 units of force.

Part A

How hard must you pull on the third rope to keep the knot from moving?


ANSWER: = 4.4

Correct units of force

Part B

In what direction must you pull on the third rope to keep the knot from moving?


ANSWER: = 83

Correct counterclockwise from the left direction

Reading Quiz 3.1

Part A

What is a vector?


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ANSWER: A quantity having both size and direction

The rate of change of velocity

A number defined by an angle and a magnitude

The difference between initial and final displacement

None of the above

Correct

Reading Quiz 3.2

Part A

What is the name of the quantity represented as ?

ANSWER: Invariant magnitude

Integral of motion

Unit vector in the x-direction

Length of the horizontal axis

Correct

Reading Quiz 3.3

Part A

This chapter shows how vectors can be added using

ANSWER: graphical addition only.

algebraic addition only.

numerical addition only.

both graphical and algebraic addition.

both graphical and numerical addition.

Correct

Conceptual Question 3.1

Part A

Can the magnitude of the displacement vector be more than the distance traveled?

ANSWER: Yes

No

Correct

Part B

Less than the distance traveled?

ANSWER: Yes

No

Correct

Part C

Explain.

ANSWER: My Answer:

Conceptual Question 3.3

Part A

If , can ?

ANSWER: Yes

No


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Correct

Part B

Can ?

ANSWER: Yes

No

Correct

Part C

For each, explain why or why not.

ANSWER: My Answer:

Conceptual Problem 3.5

Part A

Which of the following is NOT a vector?

ANSWER:

velocity

acceleration

displacement

speed

Correct

True/False Problem 3.1

Part A

The magnitude of the resultant of two vectors must be greater than or equal to the magnitude of each of these vectors.

ANSWER: True

False

Correct


Part A

The magnitude of a vector cannot be smaller than the magnitude of any of its components.

ANSWER: True

False

Correct


Part A

The components of a vector will be the same no matter what coordinate system is used to express that vector.

ANSWER: True

False

Correct


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Part A

The magnitude of the displacement vector of an object is equal to the distance that object has moved.

ANSWER: True

False

Correct

Quantitative Problem 3.2

Part A

Vector has length 5 units and directed to the north. Vector has length and is directed to the south. Calculate the magnitude and direction of

ANSWER:

11 units, north

1 units, south

11 units, south

1 units, north

Correct


Part A

Vector has length 1 units and directed to the north. Vector has length 3 units and is directed to the south. Calculate the magnitude and direction of

ANSWER:

2 units, north

4 units, north

4 units, south

2 units, south

Correct


Part A

You walk to the north, then turn 60° to your right and walk another 45 m. How far are you from where you originally started?

ANSWER:

77 m

44 m

40 m

84 m

Correct


Part A


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A rabbit trying to escape a fox runs north for 4.0 m, darts northwest for 5.0 m, then drops 1.0 m down a hole into its burrow. What is the magnitude of the net displacement of the rabbit?

ANSWER:

10 m

5.9 m

7.6 m

8.4 m

Correct

Score Summary:

Your score on this assignment is 100.7%.

You received 38.25 out of a possible total of 38 points.


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