HW0 - Introduction to MasteringPhysics + Physics Primer_
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151
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Physics
Date
Apr 3, 2024
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Pages
26
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HW0 - Introduction to MasteringPhysics + Physics Primer*
Due: 11:59pm on Friday, September 16, 2022
To understand how points are awarded, read the Grading Policy for this assignment.
A Welcome to Mastering!
Mastering presents homework items assigned by your instructor and works with you
to answer them. Homework items typically have an introduction, possibly
figures, and one or more parts for you to answer.
The purpose of the following exercises is to familiarize you with the system you will be using for the rest of your course. These exercises are not intended to
teach or test your knowledge of any specific subject material. Therefore, you will not be penalized for using hints or submitting incorrect answers.
Grading
In a graded homework item, each part counts equally toward your score on the overall item. You may
lose a fraction of the credit for a part when you submit an
incorrect answer. Whether you do lose credit and how much you lose are set by your instructor. However, you won't lose credit for most types of formatting
mistakes or for submitting a blank answer. You can find more information about how your instructor is grading you by selecting your instructor's Grading
Policy
, which can be found at the top of each assignment page. As you might expect, you will receive no credit for a part if you use the Request Answer
button. If you just can't figure out a question, there may be a way to get partial credit by using hints, as the following part will illustrate. Also, note that for
problems with hints, the option of requesting the answer is not available until you have viewed at least one hint.
Part A
What is the magic number in ? You could try to guess the magic number but you would probably
use up all your tries before getting the correct answer. Notice the View Available Hint(s)
link
underneath the question statement for this part. Selecting this link will open up a list of hints that
will guide you to the correct number.
Hint 1. How to approach the problem
Notice that there are two hints for this question. You are not required to use all of the hints or to use them in order. Each hint has a tagline that
describes its contents. Based on the tagline you can decide whether or not a particular hint will be useful to you. Your instructor may
choose to give
you a bonus for not using hints or to deduct a small penalty for using hints. If you are stuck, using the hints will usually result in a higher score than
simply trying to guess
because you may
lose fewer points for opening a hint than for getting the answer to the main question incorrect.
To help you, the magic number is equal to , where is a number between 1 and 8. Now, open up the second hint to uncover the value of .
Hint 2. What is ?
You are told from the previous hint that the magic number is equal to . For this problem, is equal to 6. Use this to compute the magic
number and enter it below.
ANSWER:
Correct
Your instructor may
choose to give you a bonus for not using hints or to deduct a small penalty for using hints. If you are stuck, using the hints will
usually result in a higher score than simply trying to guess
because you may
lose fewer points for opening a hint than for getting the answer to
the main question incorrect. Note that you are never required to use the hints; if you want to figure the question out on your own, go ahead! Upon
completing a part though, you may go back and review the hints even if you didn't use them, as you might want to when studying for a test. You
do not lose any credit for viewing hints after completing the Part.
Part B
Multiple-choice questions have a special grading rule determined by your instructor. Assume that your instructor has decided to grade these questions in
the following way: If you submit an incorrect answer to a multiple-choice question with options, you will lose of the credit for that question.
60
magic number =
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Just like the similar multiple-choice penalty on most standardized tests, this rule is necessary to prevent random guessing. If a multiple-choice question has
five answer choices and you submit one wrong answer before getting the question correct, how much credit will you lose
for that part of the question?
ANSWER:
Correct
Your instructor may choose not to deduct of the credit for a multiple-choice question with options, so refer to your instructor's
Grading Policy
on your assignment page as mentioned before to see how your instructor is grading you.
Introduction to Numeric and Symbolic Answers
Grading for Numeric Answers
Tolerance
Some numeric answers to questions need to be exact. For example, the answer to the question "How many days are in a week?" must be exactly 7.
In general though, numeric answers are graded as correct if they fall within an acceptable range (or tolerance) of the official correct answer. For example, if the
answer to a numeric problem with a tolerance of 2% was 105, then entering any value between approximately 103 and 107 would be graded as correct, as
shown in the region shaded green below. The typical grading tolerance in Mastering is 2-3%, although this value may vary (e.g. more lenient) depending on the
particular problem.
Significant Figures
While Mastering does not directly grade your answer based on the number of significant figures it contains, the tolerance mentioned above is centered around
the properly-rounded answer (as opposed to the full-precision answer). In some problems you will be explicitly told how many significant figures your answer
should contain. You may find that entering extra significant figures than required often causes your answer to still fall within the accepted tolerance and thus
graded as correct. However, there are problems where not rounding properly may cause your answer to fall outside the accepted tolerance, so it's important to
follow the general rules for significant figures, as the following example illustrates.
Suppose you are asked to find the area of a rectangle that is 3.1 wide and 4.4 long. The full-precision answer from your calculator would be 13.64 , although given the rules for significant figures, this answer should be rounded to two significant figures before being entered, or 14 . If you were to
enter 13.64 , it would not be graded as correct since this value falls outside the 2% tolerance of the properly-rounded answer of 14 . Instead, you
would receive feedback from Mastering telling you to check the rounding of your final answer, although you would not
be deducted any points.
Part A
What is the area of a rectangle that is 3.1 wide and 4.4 long? Enter the full-precision answer first to see the corresponding feedback before
entering the properly-rounded answer. (You do not need to enter the units in this case since they are provided to the right of the answer box).
ANSWER:
Correct
Calculations Involving Multiple Steps
This example raises another critical point about performing calculations in science that involve multiple steps: Make sure to keep more significant figures
than required during each intermediate step and round your final answer only as the very last step!
If you take the value displayed in your calculator and
round it before proceeding to the next step, you may encounter round-off error with your final answer deviating from the correct answer. (Tip: most calculators
have an answer button (e.g. "Ans") that allows you to use the full number from the previous calculation).
In Part B, you are asked to calculate the volume of a rectangular prism that has a length of 4.4 , a width of 3.1 , and a height of 6.3 , which is found
by multiplying the area of the base times the height. In Part A it was correct to round the result to two significant figures (14 ), since finding the area was
100
50
33
25
20
14
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the final step for that problem. However, given that this result is now being used in the calculation for Part B, you should use the full-precision value (13.64 ) and only round upon reaching the final value for the volume, which is now the final step of the calculation.
Part B
Calculate the volume of a rectangular prism with a length of 4.4 , a width of 3.1 , and a height of 6.3 . (As before, you do not need to enter the
units since they are provided to the right of the answer box.)
ANSWER:
Correct
Numeric Answers with Units
In some problems, you will be required to enter the numerical value of your answer along with the corresponding units. You can enter only numeric values
(including scientific notation), as opposed to full calculations or algebraic expressions. The same grading, tolerance, and significant figure rules apply as before
for numeric answers.
Unless the instructions explicitly ask you to use a specific unit in your answer, you are free to enter any equivalent unit within the same unit system given in the
problem (e.g. if the answer is 1 , you may also enter 100 since those quantities are equivalent). In general just be sure to use the appropriate
abbreviations and prefixes for units (for a list of recognized units in Mastering, feel free to reference Acceptable Units
).
Note that for compound units, you must use explicit multiplication between the units. For instance, newton-meter can be entered as or , but not .
Part C
The momentum of an object is determined to be 7.2
10
-3 . Express this quantity as provided or use any equivalent unit. (Note: 1 = 1000 ).
ANSWER:
Correct
Symbolic Answers
Finally, for some problems you will be required to enter a symbolic expression as your solution instead of a numeric value. The templates above the answer box
allow you to enter any special math formats or characters you may need.
In terms of grading, as long as your answer is algebraically equivalent to the correct answer, it will be graded as correct. For instance, suppose the solution to a
problem is: . You may note that this can also be rewritten as , so as a result entering this expression would also be accepted.
Part D
Enter the following expression in the answer box below: , where is the lowercase Greek letter lambda.
ANSWER:
Correct
Calculator Use
86
7.2×10
−3
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Students who successfully complete this primer will be able to:
Recognize the importance in learning the rules for the particular calculator being used
Learn specific features of calculators that will be commonly used
Employ a strategy for using a calculator that reduces the chance of making errors
Use a calculator to evaluate numerical expressions commonly encountered in physics
Note: it is strongly recommended that you work through this unit using the specific calculator
that you will use in class, on homework assignments, and for
tests!
For additional practice, you may want to review Evaluating Powers of 10
, or Solving Radical Equations
.
Overview of methods for correctly using a calculator
1. Different calculators operate in different ways. The order in which numbers, operations, and functions are entered, the markings of the keys that
perform functions, the way the information is displayed, and the way the calculator is set to interpret information entered (e.g., whether angles are
in degrees or radians), can all vary widely from calculator to calculator. Therefore, it is highly recommended that you make a habit of using the
specific calculator with which you are familiar for all of the work in your physics course.
2. The order in which information is entered into the calculator and operations are performed matters
! This is called “precedence of operation,” and
you must familiarize yourself with it. In general, you cannot simply enter an expression into a calculator in the order you read it and obtain the
correct result! A very helpful strategy for avoiding precedence of operation errors when evaluating longer expressions with a calculator is to
perform intermediate calculations one step at a time:
Select smaller portions of the whole expression.
Use the calculator to perform these intermediate evaluations.
Rewrite the expression using the intermediate results.
Proceed until the entire expression has been evaluated.
3. Below are calculator keys and operations which you must be able to use correctly for a physics class:
Clear key (you may need to use different keys to clear the display and clear values from the memory of the calculator)
Number
keys , including the decimal point Operation keys Change sign
key (note that on some calculators the change sign key looks very similar to the subtraction key; these two keys do
NOT do the same thing so you must be careful to distinguish between them)
Square root
, square
, inverse or , and exponent
(
or operations. You may need to use a shift (or
2nd
) key on the calculator for some of these operations.
Changing the angle units between degrees and radians (some calculators display the angle units currently set, others
don’t, although you should be able to change these units in your calculator’s settings)
Trigonometric functions: sine , cosine , and tangent , as well as inverse trigonometric functions: arcsine
,
or , arccosine
, or , and arctangent
, or . You may need to use a shift (or 2nd) key for some
of these operations. (Note: make sure you must use the correct setting of angle units for both trigonometric functions and inverse
trigonometric functions)
Scientific notation: you need to know the key that you use to enter numbers in scientific notation (e.g. E
, EE
, or EXP
), and the way in
which your calculator displays numbers that are expressed in scientific notation.
Other keys that may also be useful: parentheses (which can be used to group terms and reduce precedence of operation errors),
constants (such as ), memory to store intermediate results, logarithm and natural logarithm
, statistics
(
STAT
), and
more.
Learn your calculator (example 1)
Let's evaluate the quantity . You can also do this evaluation in your head. That’s an important strategy for verifying that you are
using your calculator correctly! Let’s apply the one-step-at-a-time strategy. First, perform the addition in the numerator using the calculator:
.
Then, use the calculator to perform the addition in the denominator:
.
Finally, use the calculator to perform the division to obtain the correct final numerical answer:
.
Note that if you enter the values and operations into your calculator in exactly the same order that you read them in the expression, , your
calculator will display the wrong answer for the expression. This would be a precedence of operation error! An alternative approach in a case like this is to place
parentheses around the terms in the numerator and denominator separately in your calculator: . When entered this way, your calculator,
which follows the order of operations, gives the correct answer. Now try a problem on your own.
Part A - Average acceleration
Use your calculator to evaluate .
Hint 1. Change sign vs subtraction key
Make sure you identify which calculator key is the change sign key and which key performs the subtraction operation. Verify that you are using the
change sign
key correctly by performing operations with it for which you know the answer.
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Hint 2. One-step-at-a-time
Use the one-step-at-a-time strategy (or properly utilize parentheses)
Hint 3. Verify proper use of the calculator
Verify that you are using the calculator correctly by using it with the example above and making sure you get the correct answer.
ANSWER:
Correct
When you are unsure about the operation of your calculator, verify it using operations with simple numbers for which you know the answer.
Learn your calculator (example 2)
Imagine we wish to evaluate the quantity . Start by verifying the operations of your calculator to
work with scientific notation with an expression for which you know the answer: , and then verify the squaring operation of your calculator with . Finally, once you've completed this, use your calculator to evaluate the original expression, which should come out to .
Learn your calculator (example 3)
Imagine we wish to evaluate the quantity . Start by verifying the square root operation with . On some calculators, you
enter the square root operator first and then the number (sometimes followed by a closed parenthesis), while on others you enter the number first and then the
square root operator. The same may be true for other operations as well (e.g. trigonometric and inverse trigonometric functions, inverse functions, and so on).
Make sure you learn how your calculator works with these important functions and operators! Remember, also, units are central to physics
–propagate them
into your answer
. Finally, once you've completed this, use your calculator to evaluate the original expression, which should come out to .
Try another problem now on your own.
Part B - Astronomy calculation
In analyzing distances by applying the physics of gravitational forces, an astronomer has obtained the expression Use your calculator to evaluate .
Hint 1. Verify proper use of the calculator
Make sure you verify how your calculator performs the necessary operations using it to calculate expressions for which you know the answer.
Hint 2. One-step-at-a-time
Use the one-step-at-a-time strategy
ANSWER:
Correct
When you are unsure about the operation of your calculator, verify it using operations with simple numbers for which you know the answer.
Learn your calculator (example 4)
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Imagine we wish to evaluate the quantity . Start by verifying the sine function operation and that the angle units of the
calculator are set to degrees by confirming comes out to 1. Common ways to set angle units are with a MODE
or DGR
key. On some calculators you
enter the trigonometric operator first and then the angle (sometimes followed by a closed parenthesis), while on others you enter the angle first and then the
trigonometric operator. Finally, once you've completed this, use your calculator to evaluate the original expression, which should come out to .
Learn your calculator (example 5)
Imagine we wish to evaluate the quantity . Start by verifying the arctangent function and that the angle units of the calculator are set to
degrees just as above by confirming comes out to 45°. Note that this function will likely be written as on your calculator. Finally, once you've
completed this, use your calculator to evaluate the original expression, which should come out to 60.3°, then try the last two problems.
Part C - Projectile motion
A student solving a physics problem for the range of a projectile has obtained the expression where , , and . Use your calculator to evaluate .
Hint 1. Inserting values into the equation
Write down the expression with the given values and units replacing the variables.
Hint 2. Verify proper use of the calculator
Make sure you verify how your calculator performs the necessary operations using it to calculate expressions for which you know the answer.
Hint 3. One-step-at-a-time
Use the one-step-at-a-time strategy, where one step will be to multiply by 2 before
you evaluate the sine function.
ANSWER:
Correct
When you are unsure about the operation of your calculator, verify it using operations with simple numbers for which you know the answer. Also,
check the angle units of your calculator before using trigonometric and inverse trigonometric functions.
Part D - Capstone problem
A group of physics students hypothesize that for an experiment they are performing, the speed of an object sliding down an inclined plane will be given by
the expression
.
For their experiment, , , , and . Use your calculator to obtain the value that their
hypothesis predicts for .
Hint 1. Inserting values into the equation
Write down the expression with the given values and units replacing the variables.
Hint 2. Verify proper use of the calculator
Make sure you verify how your calculator performs the necessary operations using it calculate expressions for which you know the answer.
Hint 3. One-step-at-a-time
Use the one-step-at-a-time strategy.
ANSWER:
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Correct
When you are unsure about the operation of your calculator, verify it using operations with simple numbers for which you know the answer.
Mastering Your eText
Congratulations as you begin a new course! Your electronic textbook (eText) is designed
to help you achieve your educational goals. To help you realize the full benefits of your
eText, please take a few minutes to familiarize yourself with the navigational tools and
features your eText offers.
1. Open the eText by selecting the Review link in this tutorial, in the upper right of the
screen near the scroll bar.
2. Select each icon in the menu options at the left of the eText screen to discover its
functionality.
3. Select your initials (Account settings) in the upper right corner of the screen above
the scroll bar and explore the options there.
4. When you feel comfortable navigating your eText, answer the following questions
to make sure you are ready to succeed.
Part A
As you read your eText, the menu options at the left of your screen will help you to easily navigate the eText and customize the content and your reading
experience to help you succeed in your course.
Sort each scenario to the eText icon that will help you accomplish that task. ANSWER:
Correct
Using these eText menu options will help you quickly and easily find the content you need to succeed in your course.
Part B
Reset
Help
You finish your reading and want to
view all your Pearson eTexts so you
can read for another course.
You want to find additional study tools
like videos and practice questions.
You want to view all the pages you
have previously bookmarked.
You want to get an overview of all the
topics covered in your eText at a
glance.
You want to get a sense of where you
are in your eText and to use a slider
bar to quickly move between chapters.
Class yesterday involved a debate on
the ethics and future of cloning. You
want to search for videos and other
content in your eText for more
information on this topic.
Home
Table of contents
Navigation
Search
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Sort each scenario to the eText icon that will help you accomplish that task. ANSWER:
Correct
Be sure to open and read your eText regularly. The eText is designed to give you a customized experience that helps you succeed in your course.
Part C
One nice feature of a paper book is the ability to mark a page with valuable figures, tables, or text that you plan to review later. To do this in your eText,
which icon should you select?
ANSWER:
Correct
The Bookmark icon is used to add or remove bookmarks from your eText. You can see all your bookmarks by selecting the Navigation icon.
Part D
As you encounter unfamiliar key terms in your eText, how can you learn their definitions? Note that key terms may appear in bold or in a different color in
your eText.
Select all that apply.
ANSWER:
Reset
Help
Select the Navigation icon Select the Notebook icon Select the Bookmark icon Select the Back icon You have been reading for a while and
want to rest your eyes and listen to
the next section of your chapter.
You have highlighted important
content and added notes as you read.
Now you want to compile and export
these notes as a pdf study reference.
You have finished reading the
assignment for the next quiz but want
some extra review with flashcards.
You begin reading your eText with
black letters against a white
background but want to try changing
the color mode to sepia instead.
You’ve highlighted important terms
and ideas in your eText. Now you
want to hide your highlights as you
review for an exam.
Playlist
Notebook
Study
Display settings
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Correct
There are multiple ways to learn the definitions of key terms. As you are reading your eText, you can select key terms to reveal pop-up
definitions. You can also use the Search function or navigate to the Glossary.
Scientific Notation
Students who successfully complete this primer will be able to:
Convert both very large and very small numbers into scientific notation.
Convert numbers from scientific notation into regular notation.
Multiply and divide numbers written in scientific notation.
Before working on this primer, you may need to review:
Writing values as powers of 10
For additional practice, you may want to review Conversion to Scientific Notation
.
Why write numbers in scientific notation
Understanding physics will help you describe in great detail how the world works. However, the characteristics of the world span an enormous range of
numbers. The mass of an electron is 0.000000000000000000000000000000911 (incredibly tiny!). The mass of the Sun is
1,989,000,000,000,000,000,000,000,000,000 (incredibly massive!). If you had to write, interpret, or use these numbers as written, you could very easily end
up making a minor math error. Scientific notation was invented to easily express numbers that are too large or too small to be conveniently written in decimal
form. The advantages of scientific notation are it 1) is compact, 2) helps articulate significant figures, and 3) works with numbers of any size.
The traditional format for numbers written in scientific notation is 10
is where is a number between 1 and 10 and is an integer (either positive or
negative). For the examples above, the mass of an electron can be written as 9.11
10
-31
and the mass of the Sun can be written as 1.989
10
30
.
Converting decimals to scientific notation
A day has 86,400 second . Notice that the decimal point is to the far right of the number. The first part of scientific notation is a number between 1 and 10.
Move the decimal point to the left to obtain 8.64, which equals in our traditional notation. The value for is given by the number of places the decimal point
was moved from its location in 86,400 to 8.64. Thus, 86,400 = 8.64
10
4
. Because the decimal was moved four places to the left (and the value is larger
than 1), the exponent is a positive integer. If you had a very precise clock, you could write this as 8.640
10
4
, or 8.6400
10
4
. The number of digits you
keep in the first part of the notation describes the precision of your value and determines the number of significant digits.
As another example, the fastest glacier in the world has an average speed of 0.0012 mile/hour. The first part of scientific notation is a number between 1 and
10. Move the decimal point to the right to obtain 1.2, which equals in our traditional notation. The value for is given by the number of places the decimal
point was moved from its location in 0.0012 to 1.2. Thus, 0.0012 mile/hour = 1.2
10
-3
mile/hour. Because the decimal was moved three places to the right, and
the value is smaller than 1, the exponent is a negative integer.
Part A - Write this large number in scientific notation. Determine the values of and when the following mass of the Earth is written in scientific notation: 5,970,000,000,000,000,000,000,000 .
Enter and , separated by commas.
Hint 1. Moving the decimal point
Move the decimal point to the left so you end up with a number between 1 and 10. That's the value for .
Hint 2. Finding n
Count the number of place values you moved the decimal point.
Hint 3. Sign of the exponent
For a value greater than 1, the exponent is positive.
ANSWER:
Select the Search icon and enter the term into the search field. A definition will show up in the Key terms section of the results. Select the key term in your eText and a pop-up definition will appear. Select the Playlist icon and select the module you want to hear.
Select the Table of Contents or the Navigation icon and navigate to the Glossary.
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Correct
You had to move the decimal point 24 places to the left to obtain 5.97.
Part B - Write this large number in scientific notation. Determine the values of and when the following average distance from the Sun to the Earth is written in scientific notation: 150,000,000,000 .
Enter and , separated by commas.
Hint 1. Moving the decimal point
Move the decimal point to the left so you end up with a number between 1 and 10. That's the value for .
Hint 2. Finding n
Count the number of place values you moved the decimal point.
Hint 3. Sign of the exponent
For a value greater than 1, the exponent is positive.
ANSWER:
Correct
You had to move the decimal point 11 places to the left to obtain 1.5.
Part C - Write this small number in scientific notation. Determine the values of and when the following charge of a proton is written in scientific notation: 0.000000000000000000160 .
Enter and , separated by commas.
Hint 1. Moving the decimal point
Move the decimal point to the right so you end up with a number between 1 and 10. That's the value for .
Hint 2. Finding n
Count the number of place values you moved the decimal point.
Hint 3. Sign of the exponent
For a value between 0 and 1, the exponent is negative.
ANSWER:
Correct
You had to move the decimal point 19 places to the right to obtain 1.60.
Part D - Write this small number in scientific notation. Determine the values of and when the following average magnetic field strength of the Earth is written in scientific notation: 0.0000451 .
Enter and , separated by commas.
Hint 1. Moving the decimal point
Move the decimal point to the right so you end up with a number between 1 and 10. That's the value for .
Hint 2. Finding n
Count the number of place values you moved the decimal point.
Hint 3. Sign of the exponent
=
5.97,24
=
1.5,11
=
1.60,-19
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For a value between 0 and 1, the exponent is negative.
ANSWER:
Correct
You had to move the decimal point 5 places to the right to obtain 4.51.
Rule for multiplying numbers in scientific notation
When multiplying two numbers of the form 10
, the product is: 10
10
= 10
.
Part E - Practice multiplying numbers in scientific notation
The mass of a high speed train is 4.5
10
5
, and it is traveling forward at a velocity of 8.3
10
1
. Given that momentum equals mass times velocity,
determine the values of and when the momentum of the train (in ) is written in scientific notation.
Enter and , separated by commas.
Hint 1. Rule for multiplying orders of magnitude Multiply the orders of magnitude by adding the exponents.
Hint 2. Rule for finding Don’t forget that must be between 1 and 10.
ANSWER:
Correct
Rule for dividing numbers in scientific notation
When dividing two numbers of the form 10
, the quotient is: 10
10
= 10
.
Part F - Practice dividing numbers in scientific notation
Given that average speed is distance traveled divided by time, determine the values of and when the time it takes a beam of light to get from the Sun
to the Earth (in ) is written in scientific notation. Note: the speed of light is approximately 3.0
10
8
.
Enter and , separated by commas.
Hint 1. Sun to Earth distance
Recall the average distance between the Sun and the Earth from part B.
Hint 2. Rule for finding n Divide the orders of magnitude by subtracting the exponents.
ANSWER:
Correct
Interpreting Graphs - Lines
Students who successfully complete this primer will be able to:
Identify what the different parts of the slope-intercept form of a line equation correspond to on a graph.
=
4.51,-5
=
3.7,7
=
5.0,2
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Determine what the slope of a position vs. time graph tells you about velocity.
Determine what the slope of a velocity vs. time graph tells you about acceleration.
Before working on this primer, you may need to review:
The slope-intercept form of a straight-line equation
Finding the slope of a line
The definitions for velocity and acceleration
For additional practice, you may want to review Finding the Equation (y=mx+b) from a Graph
.
Interpreting the equation of a line
The generic slope-intercept format for writing the equation of a line is where is the parameter represented by the vertical axis on a graph, is
the parameter represented by the horizontal axis on the graph, is the slope (i.e. rise/run) of the line and is the y-intercept (i.e. the value where the graph
crosses the vertical axis). For example, shows the graph of a line with the equation where the
vertical axis represents velocity (
) measured in and the horizontal axis represents time (
) measured in .
According to the generic slope-intercept format, the slope of the line is 4 and the value where the line crosses
the vertical axis is 1. However, these values have physical meaning too. Since the units on the left side are , the terms on the right side must have units of as well. If we plug in , we get ,
which tells us the velocity starts off at (also known as the initial velocity). Also, the slope must have
units of . This indicates that the slope of the line is , which represents the acceleration
.
Effect of changing parameters
In , the line equation is . Here, the line is twice
as steep, meaning the slope has doubled and the
acceleration of the object with this motion is . The
initial velocity on the other hand has not changed (
).
In , the
line
equation
is . Here, the line has the opposite slope, so the acceleration of the object with this motion is . As before, the initial velocity has not changed
(
).
In , the line equation is . Here, the line shows the slope to be , while the initial velocity is
now .
Part A - Matching a graph to a line equation
Which diagram shows the equation ?
Hint 1. Determine the slope
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Find the slope of the line by calculating the rise of the line divided by the run of the line .
Hint 2. Determine the y-intercept
The y-intercept term is the initial velocity, the location where the graph crosses the vertical axis.
Hint 3. Sign of the slope
Since the graph increases to the right, the slope is positive.
ANSWER:
Correct
The correct line has a slope of and an initial velocity of .
Part B - How changing the slope changes the graph
Which diagram shows the effect of cutting the acceleration in Part A in half and not changing the value of the initial velocity?
Hint 1. Which part of the equation represents slope
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The acceleration is represented by the slope of the line.
Hint 2. Relationship between slope and steepness
If the slope is cut in half, the graph should be half as steep.
ANSWER:
Correct
Notice how the correct line is only half as steep as in Part A.
Interpreting the slope of a velocity vs. time graph
Figure 1 shows the graph of the equation . The slope of the line is a constant value of . This allows us to sketch an acceleration vs time graph
as shown in . The equation of the line shown on this graph is . In general, the slope of the velocity vs. time graph represents the acceleration.
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We can use this slope relationship to determine the acceleration of an object if just given a velocity vs time
graph, and not the line equation(s). Consider the velocity vs time graph in . This shows an object slowing
down for two seconds, then moving at a constant rate for
two seconds, then finally speeding up for one second.
The slope of the first segment is .
The slope of the second segment is .
The slope of the third segment is .
These can be represented on the acceleration graph shown in .
Note: Don't worry about the discontinuity between line segments. In reality, velocities can't change
instantaneously so there would be a very steep line segment for a very short time interval between the flat
segments.
Part C - Finding an acceleration graph from a velocity graph
Which is the correct acceleration vs. time graph for the velocity vs. time graph shown in ?
Hint 1. What part of the graph represents acceleration?
The acceleration is represented by the slope of the line.
Hint 2. Finding the slope
Find the slope of the line by calculating the rise of the line divided by the run of the line .
ANSWER:
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Correct
You have properly applied the definition of slope to determine the acceleration of each section.
Interpreting the slope of a position vs. time graph
This same method can be used to determine the velocity vs. time graph from a simple position vs. time graph .
The equation for this graph is . If is measured in meters, each term on the right side must be in meters as
well. If we plug in , we get , which tells us the position starts off at (also known as the initial
position). Also, the slope must have units of . This indicates that the slope of the line is , which represents the
velocity. This leads to the velocity vs. time graph shown in .
We can use this slope relationship to determine the velocity of an object
if just given a position vs time graph, and not the line equation(s).
Part D - Finding a velocity graph from a position graph
Which is the correct velocity vs time graph for the position vs. time graph shown in ?
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Hint 1. What part of the graph represents velocity?
The velocity is represented by the slope of the line.
Hint 2. Finding the slope
Find the slope of the line by calculating the rise of the line divided by the run of the line .
ANSWER:
Correct
You have properly applied the definition of slope to determine the velocity of each section.
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Unit Conversions
Students who successfully complete this primer will be able to:
Identify factors associated with physically important metric prefixes.
Convert units between quantities possessing simple/compound units, multiple conversion factors, and metric prefixes.
Before working on this primer you may need to review:
Scientific notation
For additional practice, you may want to review Conversion Factors
.
Overview of unit conversion steps
1. Identify the equality relationship between the quantities that are being converted (e.g. ). Metric prefixes (see the table below)
are just a shorthand for an equality relationship between units, and are based on powers of 10 (e.g. )
2. Use the equality relationship to obtain a conversion factor
that is a ratio of the units. Since the numerator and denominator of a conversion factor
are equal, it will always have a value of exactly 1. Therefore, when you multiply a quantity by a conversion factor, it will not change its inherent
value, rather just the associated units. The conversion factor can have either
of the units in the numerator, with the other in the denominator. The
placement of the units depends on the quantity being converted
.
(e.g. or )
3. Multiply the quantity to be converted by the appropriate factor obtained in the step above. If the same unit appears in the numerator and
denominator of a fraction, it cancels. Therefore, write your conversion factor such that the unit being converted cancels out. Be sure to write these
conversion expressions with both
the numerical quantities and
the units so that they can be correctly evaluated (e.g. )
4. To convert quantities with compound units, apply multiple factors using the steps above. To check that the units being converted have all
cancelled out and only the desired units remain, you may find it helpful to cross off the cancelled units along the way (e.g.
or, for units raised to a power
)
Below is a table of important metrix prefixes:
Prefix
(abbreviation)
Power
of 10
Equality relationship
Example
Part A - Obtaining conversion factors from unit equality relationships
In an experiment you are performing, your lab partner has measured the distance a cart has traveled: . You need the distance in units of
centimeter and you know the unit equality . By which conversion factor will you multiply in order to perform the unit
conversion?
Hint 1. Which unit in the numerator and which in the denominator of the conversion factor?
Note that the quantity can be written as a fraction by writing it as . When you do this, note that the units inch are in the
numerator
(the top portion) of the fraction. Since you want to convert from inch
to a different unit, you want your conversion factor to cancel the inch
units. In order to cancel
the inch units when you mulitiply by the conversion factor, should the conversion factor have the inch units in the numerator
or in the denominator (lower portion of fraction)? ANSWER:
ANSWER:
Numerator
Denominator
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Correct
Part B - Converting compound units
You would like to know whether silicon will float in mercury and you know that can determine this based on their densities. Unfortunately, you have the
density of mercury in units of and the density of silicon in other units: . You decide to convert the density of silicon into units of
to perform the comparison. By which combination of conversion factors will you multiply to perform the unit conversion?
Hint 1. Units raised to a power?
When converting quantites that have some units raised to a power, the conversion factor for those units must be applied once for each power. Each
power of the unit must be converted.
ANSWER:
Correct
Part C - Capstone unit conversion
You have negotiated with the Omicronians for a base on the planet Omicron Persei 7. The architects working with you to plan the base need to know the
acceleration of a freely falling object at the surface of the planet in order to adequately design the structures. The Omicronians have told you that the value
is
, but your architects use the units , and from your previous experience you know that both the Omicronians and your
architects are terrible
at unit conversion. Thus, it's up to you to do the unit conversion. Fortunately, you know the unit equality relationships: and . What is the value of in the units your architects will use, in ?
ANSWER:
Correct
Solving Linear Equations
Students who successfully complete this primer will be able to:
Simplify an algebraic expression by combining like terms.
Apply the rules of algebra to isolate (i.e. solve for) a variable.
= 5.72
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Before working on this primer, you may want to review:
Working with fractions
For additional practice, you may want to review Solving Algebraic Equations
.
Many situations in physics involve solving a linear equation of the form , where and are constants and the unknown variable is raised to
the first power. To solve such an equation, the rules of algebra dictate that the same operations must be applied to both sides of such an equation, until the
variable is isolated. For instance, given the equation the first step to isolate the variable is to subtract 5 from both sides, giving
The next step is to divide both sides by 2, which isolates the variable and gives
Part A - Simplifying A Linear Equation
A linear equation will contain at least one term with the variable, and at least one constant term. Often, there are more than one of each, and the equation
must be simplified by combining like terms, such as adding and to obtain . Given the equation
which of the following choices is a correct simplified version?
Hint 1. Like Terms
Group together like terms. Combine the terms with the variable, and separately combine the constants.
Hint 2. Algebra
Recall that every operation that is applied to one side of the equation must be applied to the other side.
ANSWER:
Correct
Part B - Solving for the Variable
In Part A, we collected like terms together and simplified the equation. However, to fully solve a linear equation, the variable must be completely isolated on
one side of the equation with a coefficient of 1.
Solve the simplified equation from Part A for .
Hint 1. Isolating the Variable
What operation must we perform to both sides of the equation in order to give variable a coefficient of 1?
ANSWER:
Correct
Click here to watch a video that walks through an example on solving a linear equation, then answer the question that follows.
= -26.5
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Part C - Application: Finding Acceleration
The motion of an object undergoing constant acceleration can be modeled by the kinematic equations. One such equation is where is the final position, is the initial position, is the initial velocity, is the acceleration, and is the time.
Let's say a car starts with an initial speed of 15 , and moves between the 1000 and 5000 marks on a roadway in a time of 60 . What is its
acceleration?
Hint 1. Isolating the acceleration
You are solving for the variable in this equation, so what operations must be performed to isolate it on one side of the equation?
ANSWER:
Correct
Solving Quadratic Equations
Students who successfully complete this primer will be able to:
Recognize a quadratic equation.
Simplify a quadratic equation by combining like terms.
Apply the Quadratic Formula to find the two solutions for a quadratic equation.
Determine which solution(s) are reasonable, given the physical situation.
Before working on this primer, you may want to review:
Solving linear equations
For additional practice, you may want to review The Quadratic Formula
.
Quadratic equations occur commonly in physics, for instance when studying the motion of an object undergoing constant acceleration. They can be written in
the form:
where one term has the variable raised to the second power (i.e. squared), another has it raised to the first power, and another is a constant. Since the
highest power of in this polynomial is 2, there are two possible solutions to a quadratic equation. The equation must first be simplified and expressed in the
form above though before it can be solved using the Quadratic Formula, which is given by:
where the symbol indicates that one solution corresponds to the (
) sign and another to the (
) sign. In some cases, it is possible to solve a quadratic
equation without using the Quadratic Formula (whether it's by factoring or simply solving the equation algebraically), but in general this is not always practical.
Part A - Quadratic equations with no linear term
When a quadratic equation has no linear term (variable raised to the first power), the solution becomes simpler.
What are the two possible solutions to ?
Enter your two answers separated by a comma.
Hint 1. Obtaining a real answer
=
1.72
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In order to obtain a real answer, the number under the root cannot be negative.
Hint 2. Quadratic Formula
It is not necessary to use the Quadratic Formula to solve this particular case, but if you did, what would be the coefficients , , and , inserted
into the Quadratic Formula?
Enter the three numbers separated by commas.
ANSWER:
ANSWER:
Correct
Click here to watch a video that walks through an example on solving an equation using the quadratic formula.
To watch a supplemental video that solves for the units used in the first video, click here
.
Part B - Solving a Basic Quadratic Equation
Let's say we're given the quadratic equation . What two solutions does the Quadratic Formula give?
Enter your answers separated by a comma.
Hint 1. Coefficients
If we write the quadratic equation in the form , what would be the coefficients , , ? To express the equation in this form,
you will have to combine like terms.
Enter the three numbers separated by commas.
ANSWER:
ANSWER:
Correct
Part C - Application - Free Fall
An object moving due to gravity can be described by the motion equation , where is time, is the height at that time, is the
initial height (at ), is the initial velocity, and (the acceleration due to gravity).
If you stand at the edge of a cliff that is 75 high and throw a rock directly up into the air with a velocity of 20 , at what time will the rock hit the
ground? (Note: The Quadratic Formula will give two answers, but only one of them is reasonable.)
Express your answer in seconds.
Hint 1. Coefficients
If we plug in the quantities we are given, and write the motion equation in the form , what are the coefficients , , and ?
, , =
5,0,-15
, = 1.73,-1.73
, , =
2,8,-10
, = -5,1
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Enter the three numbers separated by commas.
ANSWER:
ANSWER:
Correct
Solving Two Equations and Two Unknowns
Students who successfully complete this primer will be able to:
Identify situations that involve solving two equations and two unknowns.
Use substitution to obtain one equation and one unknown.
Simplify the above result and solve for both of the unknowns.
Before working on this primer, you may want to review:
Solving a linear equation
Often in physics, we have a situation (such as projectile motion or the sum of forces in two dimensions) where we have two equations with the same two
unknowns in each. The most common method is substitution, which generally follows these steps:
1. Isolate one of the variables in one of the two equations. Let's say that we have
a. b. We can isolate in (a) to give .
2. Substitute this result into (b):
b. 3. Solve for the unknown that remains:
b. 4. Substitute this result into either of the original two equations to solve for the other variable:
a. Part A - Isolating a Variable
Isolating a variable in two equations is easiest when one of them has a coefficient of 1. Let's say we have the two equations
and want to isolate one of the variables, such that it appears by itself on one side of the equation. Which of the following is an equation with one of the
above variables isolated?
Hint 1. Variable coefficient
Which variable has a coefficient of 1?
ANSWER:
, , =
-4.9,20,75
= 6.45
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Correct
Part B - Substitution
Now that we have one of the variables from Part A isolated, and written in terms of the other variable, we can now substitute this into the other of the two
original equations. Which of the following options represents this?
ANSWER:
Correct
Part C - Solving for the Variables
We now have an algebraic expression with only one variable, which can be solved. Once we have that, we can plug it back into one of the original
equations (or the expression derived in Part A) to solve for the other variable. When this is done with the system of two equations from Parts A and B, what
is the solution?
Enter , then , as two numbers, separated by a comma.
Hint 1. Solving first for Let's first confirm the solution for by simplifying the previous expression, and afterwards we can proceed to solve for . What do you get when
solving for ?
ANSWER:
ANSWER:
Correct
Part D - Isolating a Variable with a Coefficient
In some cases, neither of the two equations in the system will contain a variable with a coefficient of 1, so we must take a further step to isolate it. Let's say
we now have
None of these terms has a coefficient of 1. Instead, we'll pick the variable with the smallest coefficient and isolate it. Move the term with the lowest
coefficient so that it's alone on one side of its equation, then divide by the coefficient. Which of the following expressions would result from that process?
ANSWER:
Correct
Part E - Solving for Two Variables
= 1
, =
1,-2
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Now that you have one of the two variables in Part D isolated, use substitution to solve for the two variables. You may want to review the Multiplication and
Division of Fractions and Simplifying an Expression Primers.
Enter the answer as two numbers (either fraction or decimal), separated by a comma, with first.
Hint 1. Solving first for D
As before, let's first confirm the solution for before proceeding to solve for . What do you get when solving for ? You may express your
answer as either a fraction or decimal.
ANSWER:
ANSWER:
Correct
Click here to watch a video that walks through an example on solving two equations with two unknowns, then answer the questions that follow.
Part F - Example: Finding Two Forces (Part I)
Two dimensional dynamics often involves solving for two unknown quantities in two separate equations describing the total force. The block in has a mass and is being pulled by a force on a table with coefficient of static friction . Four forces act on it:
The applied force (directed above the horizontal).
The force of gravity (directly down, where ).
The normal force (directly up).
The force of static friction (directly left, opposing any potential motion).
If we want to find the size of the force necessary to just barely overcome static friction (in
which case ), we use the condition that the sum of the forces in both directions
must be 0. Using some basic trigonometry, we can write this condition out for the forces in
both the horizontal and vertical directions, respectively, as: In order to find the magnitude of force , we have to solve a system of two equations with both and the normal force unknown. Use the methods we
have learned to find an expression for in terms of , , , and (no ).
Express your answer in terms of , , , and .
Hint 1. Isolating unknowns
Which variable are we solving for, and which should we eliminate?
ANSWER:
Correct
Part G - Example: Finding Two Forces (Part II)
For the situation in Part F, find the magnitude of the force (in ) necessary to make the block move.
= -0.57
, =
2.4,-0.57
=
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Hint 1. Numerical solution
You solved for an expression for in the last part. What numbers must be plugged in to solve for a numerical answer?
ANSWER:
Correct
Score Summary:
Your score on this assignment is 99.1%.
You received 11.89 out of a possible total of 12 points.
= 29
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