Class 090826 annotated

The idea of 'clock time':  We imagine a clock which, once it starts running, continues running.  The clock can measure the passage of time in any units (e.g., seconds, minutes, milliseconds, hours, days, years, centuries).  Times measured in one unit can of course be converted to different units.

A pendulum could constitute a 'clock', which runs in units of 'cycles' or 'half-cycles'. 

Quick Exercise:

Using the short pendulum, set up an incline so the marble will roll off the incline and fall to the floor.  Release the pendulum and start counting.  Then release the ball at the top of the incline, and count its half-cycles until it reaches the floor.  Note the count at the instant of release, at the instant it reaches the end of the ramp and at the instant it strikes the floor.

Your counts are the 'clock times' for this clock.

You observed three 'clock times' with this clock.  What were they?

Your three 'clock times' define two 'time intervals', one that lasted from release until the ball reached the end of the ramp, and another from the end of the ramp to the floor.

What were the two time intervals?

When you analyze situations involving a clock, you will need to take care to distinguish between clock times and time intervals. 

Definition of Average Rate of Change:

The average rate of change of A with respect to B is defined to be

Examples:

A child's height is 100 cm on Jan 1, 102 cm on May 1 of the same year, 105 cm on October 1 of the same year. 

To answer a question related to an average rate of change on an interval, always answer the following questions:

Answer these questions for the above example.

For a marble rolling down a ramp, off the edge and falling to the floor:

One example of a good response to this question: 

This student appears to have used a piece of notebook paper to mark landing positions, counted the lines between landing positions, then later used measurements to find the distance between the lines and converted the original data to distances in centimeters.

Assuming a change of 5 cm between the first and second slope, and recalling that the change in ramp slope was .053, we obtain

This could also be expressed as

or if we want to be very explicit about the meaning of this quantity,

For the same marble on the same ramp:

Time did not permit most of the class to get to the above observations.  No lengthy explanation is necessary in a situation like this (the instructor doesn't mind reading lengthy explanations it situations like this, but doesn't want you to spend a lot of time typing them).  A short note 'time did not permit' or 'out of time' will suffice.

Do the following, as best you can.  We've had limited discussion of graphs so if you don't do well, it's OK.  We'll have further discussion in our next class.  However do the best you can.

Describe the four graphs you have constructed (again do your best; we will soon develop some language for describing graphs).

A 'graph rectangle' is a rectangle, one of whose sides is part of the horizontal axis.

On a graph of speed in miles / hour vs. clock time in hours, we find a graph rectangle with base 3 and altitude 40.

Most students fail to fully answer this question.  The answer should be based on the definition of rate of change.  That definition leads naturally to the following sequence of questions.  If you didn't answer this question correctly, then you should make note of and begin using the following sequence of observations and questions every time you consider the meaning of a graph area:

The correct answer to the question of the area of this rectangle:

Multiplying the base by the altitude we have

area = (3 hours) * (40 miles / hour) = 120 hour * (miles / hour) = 120 miles * hour / hour = 120 miles.

On a graph of income stream in dollars per month vs. clock time in months, we find a graph rectangle with base 36 and altitude 1000. 

On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30. 

On a graph of density in grams / centimeter vs. position in centimeters, we find a graph rectangle with base 16 and altitude 50.

Question Template:  You might want to use the following template for analyzing the area of a graph rectangle.  With practice you will get used to asking these questions of yourself, and answering them correctly:

The altitudes of this graph  represent ____ in units of ____ .  This specific altitude represents a  ___ of  ____.

The base represents an interval of the horizontal axis.   ___ is measured on the horizontal axis, so this interval represents a change in ___ of  ___.

Multiplying the base by the altitude we have

The area represents the product of ___ and change in ___.  That is, we multiply the  ___, in  units of ___, by ___ in  units of ___.  This gives us the ___, in units of ___.

A 'graph trapezoid' is defined by two points on a graph, as follows:

The 'graph slope' between two points is the slope of the 'slope segment' of the graph trapezoid defined by the two points.

On a graph of speed in miles / hour vs. clock time in hours, we find graph points (2, 50) and (7, 60)

The remaining problems will be assigned again.  If you got them right, or very nearly right, you don't need to submit them again.  If you had errors, you can submit them by simply submitting a copy of your original work, with whatever changes you deem appropriate.

On a graph of income stream in dollars per month vs. clock time in months, we find the two points (16, 1000) and (40, 1200). 

On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30. 

On a graph of density in grams / centimeter vs. position in centimeters, we find the points (5, 12) and (20, 10).

 

Explain how you construct a 'graph rectangle' from a 'graph trapezoid'.

Explain how to find the area of a 'graph trapezoid'.

Ongoing question:  What is the smallest possible percent difference you think you could detect, using the pendulum, in the times required for the ball to travel down two ramps?

Drop a coin simultaneous with the release of a quarter-cycle long pendulum.  Find the minimum height at which the pendulum clearly strikes the wall first, and the maximum height at which the coin clearly strikes the floor first.

Walk down the sidewalk at constant velocity while someone times you with a pendulum of appropriate length.  Can they verify that you walked at constant velocity?

Walk down the sidewalk, increasing your velocity gradually while someone times you with a pendulum of appropriate length.  According to their results, did you speed up at a constant, an increasing or a decreasing rate?  According to your perceptions, did you speed up at a constant, an increasing or a decreasing rate?

Describe the motion of the dice on the ends of the strap, as you see them from your perspective.

What is the change in the A quantity for the interval? 100,102,105

<h3>these are the A quantities, not the changes in the A quantity</h3>

What is the change in the B quantity for the interval? 4months, 5months

What therefore is the average rate of change of A with respect to B? .5cm, .6cm

<h3>The numbers are right, but the B quantity is measured in months. So the change in the B quantity is in months. The
average rate will therefore be in cm / month, not cm.</h3>


What is the slope of your ramp when supported by a 'flat' domino? 1/30
What is the slope of your ramp when supported by a domino lying 'on its side'? 1/12
What is the slope of your ramp when supported by a domino lying 'on its end'? 1/6

How much does the slope of the ramp change when you change the domino from flat to on-its-side to on-its-end? .5
 

<h3>There are two intervals here, and you need to find the change for each interval.

Neither of the changes will be .5.

You should indicate how you got this result.</h3>

By how much does the landing position of the marble change as you move from the first slope to the second to the third?
By. 3
<h3>what is the unit of this change?</h3>

What is the average rate of change of landing position with respect to ramp slope, between the first and second slope? .4

<h3>How did you get this result and what are its units?</h3>

What do the altitudes of the graph represent? It represent the change in A

<h3>The altitudes represent the A quantity. The change in the altitude represents the change in the A quantity.

The A quantity also has a name: speed.
It also has unit: miles / hour.</h3>