If you know the initial and final values of a quantity, you find the change in that quantity by subtracting the initial from the final value.

 

If an average rate of change of one quantity with respect to another is involved, clearly identify the A and B quantities and write out the definition of the average rate in terms of these quantities.

 

If you are asked to find the change in a quantity, see if the change in that quantity is part the definition of some average rate of change.

 

<h3>You're doing well with these exercises.

However you could probably use a little more work on the last two. I'm including an expanded version of these last two problems. The original questions are still marked with &&&& at the end. The questions marked with ## break the question into smaller questions, and it is these questions you should answer. You should insert your answers to the questions ending in ## into a copy of these 'expanded' problems and submit them.</h3>

 

Questions 4-9 have each been 'expanded' into a series of specific questions.  The new questions are designated by ## at the end.

 

(Questions 1-3 involve basic definitions you are to know, and no further explanation is given, only an admonition to locate these definitions and learn them.)

 

1.  State the definition of rate of change.

 

&&&&

 

If you don't know this definition then you should thoroughly research the notes you have been provided, locate the definition and memorize it, word-for-word.

 

2.  State the definition of velocity.

 

&&&&

 

If you don't know this definition then you should thoroughly research the notes you have been provided, locate the definition and memorize it, word-for-word.

 

 

3.  State the definition of acceleration.

 

&&&&

 

If you don't know this definition then you should thoroughly research the notes you have been provided, locate the definition and memorize it, word-for-word.

 

4.  A ball rolls along a path, moving from position 20 cm to position 50 cm as its velocity increases from 5 cm/s to 15 cm/s.

 

There are four different quantities given here.  Two are velocities and two are accelerations.  How can you tell which is which?  ##

There are four different quantities given here.  Two occur at the beginning of the interval and two at the end.  How can you tell which is which?  ##

What are the units of velocity?  ##

Which of the given quantities are velocities?  ##

What is the velocity at the beginning of the interval, and what is the velocity at the end?  ##

By how much does velocity change from the beginning of the interval to the end?  ##

What are the units of position?  ##

Which of the given quantities are positions?  ##

What is the position at the beginning of the interval, and what is the position at the end?  ##

By how much does position change from the beginning of the interval to the end?  ##

5.  A ball accelerates from velocity 30 cm/s to velocity 80 cm/s during a time interval lasting 10 seconds.  Its v vs. t graph is a straight line (i.e., acceleration is uniform).

 

Explain in detail how to use the definitions you gave above to reason out

 

What are the units of velocity?  ##

What units do you get if you divide cm/s by seconds?  Do you get a unit of velocity? ##

Is it possible to get an average velocity by dividing a quantity in cm/s by seconds? ##

Explain why you don't get an average velocity if you subtract the two velocities and divide by 10 seconds. ##

Should the average velocity be closer to the initial or the final velocity, or is neither of these choices correct? ##

What is the average velocity for this interval? ##

Is the average velocity you have given closer to the initial than the final velocity? ##

Is the average velocity you have given closer to the final than the initial velocity? ##

Explain how the average velocity should be calculated using the given information. ##

and

 

What is the definition of acceleration? ##

How do you therefore calculate the acceleration? ##

What are the units of acceleration? ##

What units do you get if you multiply cm/s by seconds?  Is the result a unit of acceleration? ##

When you calculate the acceleration, your calculation involve a division, which can be written as a fraction. What is the numerator, how is it related to the definition of acceleration, what does it mean and what are its units? ##

What is the denominator, how is it related to the definition of acceleration, what does it mean and what are its units? ##

What therefore is the acceleration and what are its units? ##

 

Remember, the main goal is to use a detailed reasoning process which connects the given information to the two requested results.  You should use units with every quantity that has units, units should be included at every step of the calculation, and the algebraic details of the units calculations should be explained.

 

6.  A ‘graph trapezoid’ has ‘graph altitudes’ of 40 cm/s and 10 cm/s, and its base is 6 seconds.  Explain in detail how to find each of the following:

 

What are the altitudes of this trapezoid, what are their units and what do they mean?  ## 

What is the base of this trapezoid, what are its units and what does it mean? ##

How is the rise of the trapezoid calculated, what are the meanings of the quantities used to calculate it, and what are the units of the rise? ##

What therefore is the meaning of the rise? ##

What is the run, what are its units, and what does it mean? ##

How is the slope of the trapezoid calculated? ##

What are the units of the slope, how are they calculated from the given information, and what do they mean?##

How is the altitude of the equal-area rectangle calculated from the altitudes of the trapezoid? ##

What is the width of the equal-area rectangle?  ##

How is the area of the equal-area rectangle calculated, how are the units of the area determined from the given information, and what does the area mean? ##

What therefore is the area of the trapezoid and what does it mean? ##

 

Each calculation should include the units at every step, and the algebraic details of the units calculations should be explained.

 

7.  If the altitudes of a ‘graph trapezoid’ represent the initial and final positions of a ball rolling down an incline, in meters, and the base of the trapezoid represents the time interval between these positions in seconds, then

 

What quantities are represented by the altitudes of the trapezoid?

What are the units of the altitudes?  ## 

What quantity is represented by the base of this trapezoid, what are its units and what does it mean? ##

How is the rise of the trapezoid calculated, what are the meanings of the quantities used to calculate it, and what are the units of the rise? ##

What therefore is the meaning of the rise? ##

This question should be answered base on the meanings and the units of the rise and the run. ##

How are the units of the slope calculated from the given information? ##

What quantity is represented by the altitude of the equal-area rectangle and what are its units?  ##

What quantity is represented by the base of the equal-area rectangle and what are its units?  ##

How are the altitude and base of the equal-area rectangle used to calculate the area of the trapezoid? ##

How are the units of the area calculated from the given information? ##

Each answer should include a complete explanation, reasoned out from the geometry of the trapezoid and the definitions you gave at the beginning.

 

8.  If the altitudes of a ‘graph trapezoid’ represent the initial and final velocities of a ball rolling down an incline, in meters / second, and the base of the trapezoid represents the time interval between these velocities in seconds, then

 

What quantities are represented by the altitudes of the trapezoid? ##

What are the units of the altitudes?  ## 

What quantity is represented by the base of this trapezoid, what are its units and what does it mean? ##

How is the rise of the trapezoid calculated, what are the meanings of the quantities used to calculate it, and what are the units of the rise? ##

What therefore is the meaning of the rise? ##

This question should be answered based on the meanings and the units of the rise and the run. ##

How are the units of the slope calculated from the given information? ##

What quantity is represented by the altitude of the equal-area rectangle and what are its units?  ##

What quantity is represented by the base of the equal-area rectangle and what are its units?  ##

How are the altitude and base of the equal-area rectangle used to calculate the area of the trapezoid? ##

How are the units of the area calculated from the given information? ##

Each answer should include a complete explanation, reasoned out from the geometry of the trapezoid and the definitions you gave at the beginning.

 

9.  A ball rolls along a path, moving from position 20 cm to position 50 cm as its velocity increases from 5 cm/s to 15 cm/s.

 

What is the ball's change in position?  ##

What is its average velocity?  ##

What is the definition of average velocity?##

How therefore do you calculate the time interval when you know the change in position and the average velocity? ##

What is the change in the ball's velocity? ##

What is the definition of average acceleration? ##

How do you therefore calculate the average acceleration from the information you now have? ##