#$&*

course PHY 201

6/22 10:30

003. Velocity Relationships*********************************************

Question: `q001. Note that there are 13 questions in this assignment.

vAve = `ds / `dt, which is the definition of average velocity and which fits well with our intuition about this concept.

If displacement `ds is measured in meters and the time interval `dt is measured in seconds, in what units will vAve be obtained?

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Your solution:

The units obtained for vAve will be meters per second m/sec.

confidence rating #$&*:

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Given Solution:

vAve = `ds / `dt.

The units of `ds are m and the units of `dt are sec, so the units of `ds / `dt must be m / sec.

Thus vAve is in m/s.

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q002. If the equation vAve = `ds / `dt is to be solved for `ds we multiply both sides of the equation by `dt to obtain `ds = vAve * `dt. If vAve is measured in cm / sec and `dt in sec, then in what units must `ds be measured?

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Your solution:

The units measured would be cm/sec * sec if solving for `ds.

confidence rating #$&*:

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Given Solution:

Since vAve is in cm/sec and `dt in sec, `ds = vAve * `dt must be in units of cm / sec * sec = cm.

STUDENT QUESTION

I don’t get how sec and sec would cancel each other out

INSTRUCTOR RESPONSE

cm / s * s means

(cm/s) * s, which is the same as

(cm / s) * (s / 1). Multiplying numerators and denominators we have

(cm * s) / (s * 1) or just

(cm * s) / s, which is the same as

cm * (s / s) = cm * 1 = cm.

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q003. Explain the algebra of multiplying the unit cm / sec by the unit sec.

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Your solution:

We multiply cm/sec by sec we are multiplying the fractions cm / sec and sec / 1. We get cm * sec / ( sec * 1). This equals cm.

confidence rating #$&*:

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Given Solution:

When we multiply cm/sec by sec we are multiplying the fractions cm / sec and sec / 1. When we multiply fractions we will multiply numerators and denominators. We obtain cm * sec / ( sec * 1). This can be rearranged as (sec / sec) * (cm / 1), which is the same as 1 * cm / 1. Since multiplication or division by 1 doesn't change a quantity, this is just equal to cm.

STUDENT RESPONSE:

For some reason this question just isn't making sense to me.

INSTRUCTOR RESPONSE:

In a self-critique you need to address the given solution in detail. A general statement such as yours gives me no information on what you understand. I need this information as a basis for helping you with what you don't understand.

In order to give me the information I need you should be addressing each statement, and each phrase, to show me what you do and do not understand.

The given solution can be broken into individual statements:

1. When we multiply cm/sec by sec we are multiplying the fractions cm / sec and sec / 1.

2. When we multiply fractions we will multiply numerators and denominators.

3. We obtain cm * sec / ( sec * 1).

4. This can be rearranged as (sec / sec) * (cm / 1), which is the same as 1 * cm / 1. Since multiplication or division by 1 doesn't change a quantity, this is just equal to cm.

Do you understand Statement 1?

If not, have you written out the expressions cm/sec and sec/1 in standard form? (You might want to review the link given at the end of the Typewriter Notation exercise from Orientation, which should be posted at your access page).

As best you can communicate it, what do you and do you not understand about this statement?

Do you understand Statement 2? If not, what do you and do you not understand about this statement?

Do you understand Statement 3? If not, have you written out the multiplication of cm/sec and sec/1 on paper? The multiplication is (cm / sec) * (sec / 1). Again, if you aren't sure how to write this out, refer to the link at the end of the Typewriter Notation exercise.

Do you understand Statemet 4? If not do you understand that (sec / sec) * (cm / 1) is equal to sec * cm / (sec * 1), which is in turn equal to (cm * sec) / (sec * 1)? If not, specifically what do you and do you not understand?

If you don't understand anything, then you should start with a review of basic fractions, a topic which is very much neglected in the typical curriculum in U.S. schools. Then you should return to these questions and give your best answers.

A good link, current as of Sept. 2010:

http://www.themathpage.com/arith/multiply-fractions-divide-fractions.htm

You should submit a copy of question `q003, your solution, the given solution and this note. Insert your answers and/or additional specific questions and mark with &&&& before and after each insertion, then submit using the Submit Work Form.

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q004. If the definition vAve = `ds / `dt is to be solved for `dt we multiply both sides of the equation by `dt to obtain vAve * `dt = `ds, then divide both sides by vAve to get `dt = `ds / vAve. If vAve is measured in km / sec and `ds in km, then in what units must `dt be measured?

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Your solution:

dt would be measured in seconds. km / km/sec so that the km cancel each other out and leave seconds.

confidence rating #$&*:

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Given Solution:

Since `dt = `ds / vAve and `ds is in km and vAve in km/sec, `ds / vAve will be in km / (km / sec) = seconds.

STUDENT SOLUTION LACKING DOCUMENTATION

seconds

INSTRUCTOR RESPONSE

You should show the reasoning; we know in advance that `dt will be in seconds, but be sure you understand how to get there from the given units.

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Self-critique (if necessary): ok

Self-critique rating: ok

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Question: `q005. Explain the algebra of dividing the unit km / sec into the unit km.

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Your solution:

Multiplying numerators and denominators we get (km * sec) / (1 * km). (km / km) * (sec / 1)= seconds.

confidence rating #$&*:

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Given Solution:

The division is km / (km / sec). Since division by a fraction is multiplication by the reciprocal of the fraction, we have km * (sec / km). This is equivalent to multiplication of fractions (km / 1) * (sec / km). Multiplying numerators and denominators we get (km * sec) / (1 * km), which can be rearranged to give us (km / km) * (sec / 1), or 1 * sec / 1, or just sec.

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q006. If an object moves from position s = 4 meters to position s = 10 meters between clock times t = 2 seconds and t = 5 seconds, then at what average rate is the position of the object changing (i.e., what is the average velocity of the object) during this time interval? What is the change `ds in position, what is the change `dt in clock time, and how do we combine these quantities to obtain the average velocity?

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Your solution:

‘ds= 6m

‘dt=3sec

6meters/3seconds = vAve = 2meters/sec

confidence rating #$&*:

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Given Solution:

We see that the changes in position and clock time our `ds = 10 meters - 4 meters = 6 meters and `dt = 5 seconds - 2 seconds = 3 seconds. We see also that the average velocity is vAve = `ds / `dt = 6 meters / (3 seconds) = 2 meters / second. Comment on any discrepancy between this reasoning and your reasoning.

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Self-critique (if necessary): ok

Self-critique rating: ok

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Question: `q007. Symbolize this process: If an object moves from position s = s1 to position s = s2 between clock times t = t1 and t = t2, then what expression represents the change `ds in position and what expression represents the change `dt in the clock time?

What expression therefore symbolizes the average velocity between the two clock times.

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Your solution:

`ds = s2 - s1 and `dt = t2 - t1

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Given Solution:

The change in position is `ds = s2 - s1, obtained as usual by subtracting the first position from the second. Similarly the change in clock time is `dt = t2 - t1.

The symbolic expression for the average velocity is therefore

•vAve = `ds / `dt = (s2 - s1) / (t2 - t1).

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q008. On a graph of position s vs. clock time t we see that the first position s = 4 meters occurs at clock time t = 2 seconds, which corresponds to the point (2 sec, 4 meters) on the graph, while the second position s = 10 meters occurs at clock time t = 5 seconds and therefore corresponds to the point (5 sec, 10 meters). If a right triangle is drawn between these points on the graph, with the sides of the triangle parallel to the s and t axes, the rise of the triangle is the quantity represented by its vertical side and the run is the quantity represented by its horizontal side. This slope of the triangle is defined as the ratio rise / run. What is the rise of the triangle (i.e., the length of the vertical side) and what quantity does the rise represent? What is the run of the triangle and what does it represent?

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Your solution:

The rise of the triangle represents the change in the position. From the first point to the second is 10 m - 4 m = 6 m. The run of the triangle represents the change in the clock time. 5 s - 2 s = 3 s.

Confidence rating: 2

Given Solution:

The rise of the triangle represents the change in the position coordinate, which from the first point to the second is 10 m - 4 m = 6 m. The run of the triangle represents the change in the clock time coordinate, which is 5 s - 2 s = 3 s.

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q009. What is the slope of this triangle and what does it represent?

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Your solution:

6 meters / 3 seconds = 2 meters / second

confidence rating #$&*:

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Given Solution:

The slope of this graph is 6 meters / 3 seconds = 2 meters / second.

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q010. In what sense does the slope of any graph of position vs. clock time represent the velocity of the object? For example, why does a greater slope imply greater velocity?

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Your solution:

The slope represents rise / run or average velocity.

confidence rating #$&*:

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Given Solution:

Since the rise between two points on a graph of velocity vs. clock time represents the change `ds in position, and since the run represents the change `dt clock time, the slope represents rise / run, or change in position / change in clock time, which is `ds / `dt. This is equal to the average rate of change of position with respect to clock time, which is the definition of average velocity.

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Self-critique (if necessary): ok

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Self-critique rating: ok

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Question: `q011. As a car rolls from rest down a hill, its velocity increases. Describe a graph of the position of the car vs. clock time.

If you have not already done so, tell whether the graph is increasing at an increasing rate, increasing at a decreasing rate, decreasing at an increasing rate, decreasing at a decreasing rate, increasing at a constant rate or decreasing at a constant rate.

Is the slope of your graph increasing or decreasing?

How does the behavior of the slope of your graph indicate the condition of the problem, namely that the velocity is increasing?

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Your solution:

The graph would be increasing at a increasing rate since distance is increasing and the time required is increasing. As the cars position changes the time interval at which it change would be getting less so the graph would be curving upward. The slope of the graph would be increasing. velocity is increasing because the distance between time and position change is narrowing.

confidence rating #$&*:

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Given Solution:

The graph should have been increasing, since the position of the car increases with time (the car gets further and further from its starting point). The slope of the graph should have been increasing, since it is the slope of the graph that indicates velocity. An increasing graph within increasing slope is said to be increasing at an increasing rate (an alternative description would be that the graph is increasing and concave up).

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Self-critique (if necessary): ok

Self-critique rating: ok

If you understand the assignment and were able to solve the previously given problems from your worksheets, you should be able to complete most of the following problems quickly and easily. If you experience difficulty with some of these problems, you will be given notes and we will work to resolve difficulties.

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Question: `q012. If at clock time t = t_1 the position of an object is x = x_1, while at clock time t = t_2 its position is x = x_2, then what is its average velocity during the corresponding interval?

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Your solution:

X_2-X_1/ T_2-T_1

confidence rating #$&*:

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Question: `q013. On a graph of position x vs. clock time t, what is the average slope between the point (t_1, x_1) and (t_2, x_2)? What is the meaning of this average slope, and why?

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Your solution:

Slope=X2-X1/T2-T1

confidence rating #$&*:

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Self-critique rating: ok

@&

That is the expression for the slope, though numerator and denominator each really need to be enclosed in parenthses because of order of operations.

However you haven't specified the meaning of that slope for the graph in question.

*@

You should submit the above questions, along with your answers and self-critiques. You may then begin work on the Questions, Problem and Exercises, as instructed below.

Questions, Problems and Exercises

You should answer the questions and work the problems as given below, as appropriate to your course. Your work should normally be handwritten, should include diagrams and sketches where appropriate, and should go into your Physics notebook (not into your lab notebook).

If the course is not specified for a problem, then students in all physics courses should do that problem.

Principles of Physics students need not do the questions or problems that are specified for General College Physics or University Physics.

General College Physics students need not do questions or problems specified for University Physics.

University Physics students should do all questions and problems.

Principles of Physics students may if they wish do some of the questions and problems specified for General College Physics, but this is neither expected nor required. These problems are accessible to Principles of Physics students, but are generally more challenging that what is expected of students in this course. (Some University Physics problems will also be accessible to Principles of Physics students, though some will not.)

General College Physics students who wish to do so are welcome to work some or all of the University Physics questions and problems, though this is neither expected nor required. Many of the University Physics questions and problems are more challenging than those expected of General College Physics students, and a number of the problems require the use of calculus, which is not expected of General College Physics students.

You are not expected to submit these questions and problems. It would take too much of your time to key in all the answers and solutions. You should solve these problems and answer these questions in your notebook, in a form you can later reference and, if you later desire, revise. The Query at the end of the assignment will ask you selected questions, which you will at that time be expected to answer based on the work you have done in your notebook.

Remember that you are always welcome to ask questions at any time. Any question about a problem should include a copy of the problem and a summary of what you do and do not understand about it.

Questions related to q_a_

1. If we were to multiply the unit cm / sec by the unit sec / hr what unit would we get?

2. If we were to divide the unit cm by the unit cm / sec what unit would we get?

3. In this qa we began with the definition of average velocity. We have seen that this definition is consistent with our idea of speed as distance traveled divided by time required. However our definition of velocity can produce a negative result, which was not the case for speed. The negative result will, for example, occur when the change in position is negative while the change in clock time is positive.

•How is it that the change in position can be negative?

4. Having defined average velocity, we used symbols to stand for the initial and final positions and clock times. What expression did this give us for the average velocity?

5. Having defined average velocity, we then represented a pair of initial and final positions and clock times on a graph of position vs. clock time. What did we learn about the graph?

6. What are some possible units for position? What are some possible units for clock time? What therefore are some possible units for rate of change of position with respect to clock time?

Questions related to text

1. What is 3.2 km + 340 meters, to the correct number of significant figures?

2. Find 1.80 m + 142.5 cm + 5.34 * 10^5 micro m, to the correct number of significant figures. You should know that 'micro' indicates multiplication by 10^-6, so a micro m is 10^-6 meter.

3. Find 3.84 kilograms - 3842 grams, to the appropriate number of significant figures.

Questions related to key systems

1. Pendulums of lengths 20 cm and 25 cm are counted for one minute. The counts are respectively 69 and 61. To how many significant figures do we know the difference between these counts?

Questions/problems for University Physics Students

1. Find the x and y components of two displacements, one of 4.0 meters at 60 degrees and the other of 3.0 meters at 135 degrees. Both angles are measured in the counterclockwise direction from the positive x axis.

2. If you drive 2.6 km north, then 4 km east before turning 45 deg to your left and traveling an addition 3.1 km, then what is the magnitude and direction of your displacement from the original position to the final position? Sketch a figure depicting this motion, describe the figure and describe how it appears to support your result.

3. What do you think is the uncertainty in the magnitude of a vector whose components are given as 0.7 cm in the x direction and 5.932 cm in the y direction? What do you think is the uncertainty in its direction?

"

Self-critique (if necessary):

------------------------------------------------

Self-critique rating:

You should submit the above questions, along with your answers and self-critiques. You may then begin work on the Questions, Problem and Exercises, as instructed below.

Questions, Problems and Exercises

You should answer the questions and work the problems as given below, as appropriate to your course. Your work should normally be handwritten, should include diagrams and sketches where appropriate, and should go into your Physics notebook (not into your lab notebook).

If the course is not specified for a problem, then students in all physics courses should do that problem.

Principles of Physics students need not do the questions or problems that are specified for General College Physics or University Physics.

General College Physics students need not do questions or problems specified for University Physics.

University Physics students should do all questions and problems.

Principles of Physics students may if they wish do some of the questions and problems specified for General College Physics, but this is neither expected nor required. These problems are accessible to Principles of Physics students, but are generally more challenging that what is expected of students in this course. (Some University Physics problems will also be accessible to Principles of Physics students, though some will not.)

General College Physics students who wish to do so are welcome to work some or all of the University Physics questions and problems, though this is neither expected nor required. Many of the University Physics questions and problems are more challenging than those expected of General College Physics students, and a number of the problems require the use of calculus, which is not expected of General College Physics students.

You are not expected to submit these questions and problems. It would take too much of your time to key in all the answers and solutions. You should solve these problems and answer these questions in your notebook, in a form you can later reference and, if you later desire, revise. The Query at the end of the assignment will ask you selected questions, which you will at that time be expected to answer based on the work you have done in your notebook.

Remember that you are always welcome to ask questions at any time. Any question about a problem should include a copy of the problem and a summary of what you do and do not understand about it.

Questions related to q_a_

1. If we were to multiply the unit cm / sec by the unit sec / hr what unit would we get?

2. If we were to divide the unit cm by the unit cm / sec what unit would we get?

3. In this qa we began with the definition of average velocity. We have seen that this definition is consistent with our idea of speed as distance traveled divided by time required. However our definition of velocity can produce a negative result, which was not the case for speed. The negative result will, for example, occur when the change in position is negative while the change in clock time is positive.

•How is it that the change in position can be negative?

4. Having defined average velocity, we used symbols to stand for the initial and final positions and clock times. What expression did this give us for the average velocity?

5. Having defined average velocity, we then represented a pair of initial and final positions and clock times on a graph of position vs. clock time. What did we learn about the graph?

6. What are some possible units for position? What are some possible units for clock time? What therefore are some possible units for rate of change of position with respect to clock time?

Questions related to text

1. What is 3.2 km + 340 meters, to the correct number of significant figures?

2. Find 1.80 m + 142.5 cm + 5.34 * 10^5 micro m, to the correct number of significant figures. You should know that 'micro' indicates multiplication by 10^-6, so a micro m is 10^-6 meter.

3. Find 3.84 kilograms - 3842 grams, to the appropriate number of significant figures.

Questions related to key systems

1. Pendulums of lengths 20 cm and 25 cm are counted for one minute. The counts are respectively 69 and 61. To how many significant figures do we know the difference between these counts?

Questions/problems for University Physics Students

1. Find the x and y components of two displacements, one of 4.0 meters at 60 degrees and the other of 3.0 meters at 135 degrees. Both angles are measured in the counterclockwise direction from the positive x axis.

2. If you drive 2.6 km north, then 4 km east before turning 45 deg to your left and traveling an addition 3.1 km, then what is the magnitude and direction of your displacement from the original position to the final position? Sketch a figure depicting this motion, describe the figure and describe how it appears to support your result.

3. What do you think is the uncertainty in the magnitude of a vector whose components are given as 0.7 cm in the x direction and 5.932 cm in the y direction? What do you think is the uncertainty in its direction?

"

Self-critique (if necessary):

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Self-critique rating:

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