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The problem:
Here is the definition of rate of change of one quantity with respect to another:
The average rate of change of A with respect to B on an interval is
average rate of change of A with respect to B = (change in A) / (change in B)
Apply the above definition of average rate of change of A with respect to B to each of the following. Be sure to identify the quantity A, the quantity B and the requested average rate.
If the position of a ball rolling along a track changes from 10 cm to 20 cm while the clock time changes from 4 seconds to 9 seconds, what is the average rate of change of its position with respect to clock time during this interval?
answer/question/discussion: The average rate of change for the rolling ball from its position and respect to clock time is 2 cm/s. Quanity A(20cm-10cm=10cm)
20 cm/s - 10 cm/s is 10 cm/s.
Quantities having the same units are like terms and their sum or difference is also a like term.
Formally the cm/s 'factor out' so 20 cm/s - 10 cm/s = (20 - 10) cm/s = 10 cm/s.
Quanity B(9sec-4sec=5sec)
Average Rate of Change(10cm/5sec=2cm/s)
10 cm/sec / (5sec) = 2 (cm / s) / s = 2 cm/s * 1/s = 2 cm/s^2, or 2 cm/s/s (two cm per second, per second)
If the velocity of a ball rolling along a track changes from 10 cm / second to 40 cm / second during an interval during which the clock time changes by 3 seconds, then what is the average rate of change of its velocity with respect to clock time during this interval?
answer/question/discussion: During this interval, the average rate of change in the ball's velocity with respect
to clock time is 10cm/s. A= (40cm/s-10cm/s=30cm/s)
B= 3 seconds
C= Rate= (30cm/s)/3sec=10cm/s
30 cm/s / (3 s) = 10 cm/s / s = 10 cm/s * 1/s = 10 cm/s^2.
If the average rate at which position changes with respect to clock time is 5 cm / second, and if the clock time changes by 10 seconds, by how much does the position change?
answer/question/discussion: The average position change was 50cm. A=5cm/s, B=10sec
Change=5cm/sx10sec=50cm
This is the position change, not the average position change. There is more than one velocity associated with this interval, but only one change in position. So we can speak of an average velocity, or of average rate of change of velocity with respect to clock time, but not of average change in position.
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20 min
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Good, though you did make a couple of errors. See my notes and let me know if you have questions.