course Phy 201
The average rate of change of A with respect to B is defined to bexxxx
average rate = change in A / change in B
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.
What was the clock time at each measurement?
Answer:
Jan 1 May 1 Oct 1
0 4 9
Clock time:
What are the changes in clock times between measurements?
Answer:
From Jan 1 to May 1 the change in clock time (months) was 4,
From May 1 to Oct 1 the clock time change was 5,
And from Jan 1 to Oct 1 the clock time changed by 9
At what average rate did the child's height change with respect to clock time between Jan 1 and May 1?
Answer:
A is the childs height (cm)
B is the clock time (in months)
Ave. rate = Change in A / Change in B = 2 / 4 = ½ cm/month
At what average rate did the child's height change with respect to clock time between May 1 and October 1?
Answer:
A is the childs height (cm)
B is the clock time (in months)
Ave. rate = Change in A / Change in B = 5/9 cm / month
To answer a question related to an average rate of change on an interval, always answer the following questions:
What is the A quantity?
What is the B quantity?
What is the change in the A quantity for the interval?
What is the change in the B quantity for the interval?
What therefore is the average rate of change of A with respect to B?
Answer these questions for the above example.
For a marble rolling down a ramp, off the edge and falling to the floor:
What is the slope of your ramp when supported by a 'flat' domino?
Answer:
The domino is 1 cm thick so the height of the ramp is 1 cm, the ramp is 30 cm long (or close enough). Slope is the change in height / the change of length (rise / run).
A is 1 cm, B is 30 cm therefore the slope is A /B = 1/30 cm
What is the slope of your ramp when supported by a domino lying 'on its side'?
Answer:
The domino is 2.5 cm wide and the ramp is the same (30 cm)
A = 2.5 cm, B = 30 cm, slope = 2.5 / 30 = 1/12 cm
What is the slope of your ramp when supported by a domino lying 'on its end'?
Answer:
The domino is 5 cm long and the ramp is 30 cm long
A = 5 cm, B = 30 cm, slope = 5 / 30 = 1/6 cm
How much does the slope of the ramp change when you change the domino from flat to on-its-side to on-its-end?
Answer:
When we changed the domino from flat to on its side the slope increased from 1/30 cm to 1/12 cm so the slope is steeper when the domino is in a higher position.
When we changed the domino form its side to on its end the slope again increased from 1/12 cm to 1/6 the slope doubled because the height doubled.
By how much does the landing position of the marble change as you move from the first slope to the second to the third?
Answer:
From the first slope the marble hit about 8 cm farther on the second slope
From the second slope the marble hit 11 cm farther on the third slope.
What is the average rate of change of landing position with respect to ramp slope, between the first and second slope?
Answer:
A is landing position
B is slope
A=8 and B= 1.5/30 8/ (1.5/30) = 160
Those are the changes in the A and B quantities, not the A and B quantities themselves. Be precise about the wording.
Units are also needed. The A quantity has units of cm; the B quantity is a unitless slope (the rise and run of the ramp both have the same units, so when you divide the units divide out).
The average rate of change would be 160 cm, or if you prefer 160 cm / unit of ramp slope.
What is the average rate of change of landing position with respect to ramp slope, between the second and third slope?
Answer:
A=3 B= 2.5/30
3/ (2.5/30) = 36
For the same marble on the same ramp:
How long does it take the ball to roll down the incline with the domino lying 'flat'?
Answer:
It takes close to 2 oscillations of the book size pendulum about 2.264 seconds
How long does it take the ball to roll down the incline with the domino lying 'on its side'?
How long does it take the ball to roll down the incline with the domino lying 'on its end'?
For each interval, what is the average rate of change of the time required to roll down the incline with respect to ramp slope?
For each interval, what is the average rate of change of the ball's position with respect to clock time as it rolls down the ramp?
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.
Make a graph of marble position vs. clock time as it rolls down an incline of length 30 cm, starting from rest, in 3 seconds.
Make a graph of marble velocity vs. clock time as it rolls down an incline of length 30 cm, starting from rest, in 3 seconds.
For your marble rolling down the three inclines, graph position vs. clock time for each incline.
For your marble rolling down the three inclines, graph velocity vs. clock time for each incline.
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.
The quantity which is represented by the length of the side which is part of the horizontal axis is the 'base' of the graph rectangle.
The quantity represented by the length of either of the sides perpendicular to the 'base' is the 'altitude' of the graph rectangle.
The 'area' of the graph rectangle is the product of the quantity represented by its 'base' and the quantity represented by its 'altitude'.
On a graph of speed in miles / hour vs. clock time in hours, we find a graph rectangle with base 3 and altitude 40.
What does the altitude of the graph represent?
Answer:
Mph, speed
What does the base of the graph represent?
Answer:
Hours, clock time
What is the area of the graph?
Answer:
120 miles, because 3 hr. / 40 mi/hr = 120 mi
3 hr. / (40 mi/hr) = .075 hr^2 / mi.
3 hr. * 40 mi/hr = 120 mi, which is correct and is what you intended.
What does the area of the graph represent?
Answer:
Distance traveled (in 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.
What does the altitude of the graph represent?
Answer:
dollars
What does the base of the graph represent?
Answer:
months
What is the area of the graph?
Answer:
36,000 dollars
What does the area of the graph represent?
Answer:
Profit over 36 months
Good, but technically profit is income - expenses, and the area of this graph represents only income as opposed to profit.
On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30.
What does the altitude of the graph represent?
Answer:
Force (pounds)
What does the base of the graph represent?
Answer:
Position (feet)
What is the area of the graph?
Answer:
6,000 pounds
200 pounds * 30 feet = 6 000 ft * pounds, not 6 000 pounds.
What does the area of the graph mean?
Answer:
6,000 total pounds on 30 feet
On a graph of density in grams / centimeter vs. position in centimeters, we find a graph rectangle with base 16 and altitude 50.
What does the altitude of the graph represent?
Answer:
Grams per centimeter
What does the base of the graph represent?
Answer:
Position
What is the area of the graph?
Answer:
800 grams
What does the area of the graph mean?
Answer:
Total grams
A 'graph trapezoid' is defined by two points on a graph, as follows:
The 'left altitude' is the line segment parallel to the y axis, running from the leftmost of the two points to the horizontal axis.
The 'right altitude' is the line segment parallel to the y axis, running from the rightmost of the two points to the horizontal axis.
The 'slope segment' is the line segment between the two points.
The 'base' is the part of the x axis between the two altitudes.
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)
What do the altitudes of the graph represent?
Speed (mph)
What is the rise between the two points of this graph?
Answer:
Clock time (hours)
What is the run between these points?
Answer:
5 hours
What therefore is the slope associated with this graph trapezoid?
Answer:
2
What does the slope mean?
Answer:
For every hour the speed increases by 2 mph
What does the base of the graph represent?
Answer:
Clock time (hours)
What are the dimensions of the equal-area graph rectangle?
Answer:
Base- 5 hours, altitude- 55 mph
What is the area of the graph?
Answer:
275 miles
What does the area of the graph represent?
Answer:
Total distance in miles
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).
What do the altitudes of the graph represent?
Answer:
Income in dollars
What is the rise between the two points of this graph?
Answer:
200 dollars
What is the run between these points?
Answer:
24
What therefore is the slope associated with this graph trapezoid?
Answer:
M= change in alt/ change in base, therefore m= 25/3
What does the slope mean?
Answer:
The company is gaining 25 dollars of profit every three months.
What does the base of the graph represent?
Answer:
Clock time (months)
What are the dimensions of the equal-area graph rectangle?
Base of 24 months, alt of 1100 dollars
What is the area of the graph?
Answer
26,400 dollars
What does the area of the graph represent?
Answer:
Total profit in dollars
On a graph of force in pounds vs. position in feet, we find a graph rectangle with base 200 and altitude 30.
What do the altitudes of the graph represent?
Answer:
Force in pounds
What is the rise between the two points of this graph?
Answer:
There is no change
What is the run between these points?
Answer:
No run
What therefore is the slope associated with this graph trapezoid?
This a rectangular graph so the slope is 0
What does the slope mean?
Answer:
There is no change in the rise.
What does the base of the graph represent?
Position in feet
What are the dimensions of the equal-area graph rectangle?
The same, base 200 and alt 30
What is the area of the graph?
200 * 30 = 6,000
What does the area of the graph represent?
Answer:
Total pounds for 30 feet
On a graph of density in grams / centimeter vs. position in centimeters, we find the points (5, 12) and (20, 10).
What do the altitudes of the graph represent?
Answer:
Density (g/cm)
What is the rise between the two points of this graph?
Answer:
-2
What is the run between these points?
Answer:
15
What therefore is the slope associated with this graph trapezoid?
Answer:
-2/15
What does the slope mean?
Answer:
The density is decreasing
What does the base of the graph represent?
Answer:
Position (cm)
What are the dimensions of the equal-area graph rectangle?
Answer:
Base 15 cm, Alt. 11 g/cm
What is the area of the graph?
Answer:
165 g
What does the area of the graph represent?
Answer:
Total grams
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?
Answer:
About 25 %
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?
Answer:
Yes, if you kept the right pace
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?
Answer:
Constant, with the pendulum ticks
According to your perceptions, did you speed up at a constant, an increasing or a decreasing rate?
Constant
Describe the motion of the dice on the ends of the strap, as you see them from your perspective
Overall you're in good shape.
I haven't commented on all your answers, but have commented on a sufficient number to help you clarify a couple of points.
I'll be sending a complete commentary on this homework just before Monday's class, and we'll also talk more about these ideas in class..