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Submitting Assignment: Uniformly Accelerated Motion Scheme
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Definition 1: Average velocity is an average rate of change of position with respect to clock time.
Definition 2: Average acceleration is an average rate of change of velocity with respect to clock time.
Rule 1:The definition of an average rate of change of a quantity A with respect to another quantity B is
The following will be used extensively and is worth noting:
From this it follows that
and also that
Definition of change in a quantity: The change in a quantity is equal to the final value of that quantity minus the initial value of that quantity.
Now, suppose an object accelerates from initial velocity 5 cm/s at clock time t = 12 seconds to final velocity 25 cm/s at t = 30 seconds.
The above definition can be applied to this information to find the average rate of change of velocity with respect to clock time.
You may use this program to with these quantities, or you may follow the process using different quantities. If you are using different quantities, please state the quantities you will be using and, if applicable, state the context in which these quantities appear. If you are using the quantities specified above, you may leave this box blank.
To apply the definition you need to follow these steps:
This is a 7-step process. In the box below perform this process, one step to a line. Starting in the 8th line, briefly restate the process in your own words and explain why this process makes sense in terms of the example.
In the preceding example:
Answer these questions in the box below in the first two lines. Starting in the third line explain how you were able to identify the initial and final velocity and the initial and final clock time.
Average velocity is average rate of change of position with respect to clock time. When determining average rate of change of velocity with respect to clock time using the above definition, the quantities A and B had to be velocity and position.
Answer these questions in the first two lines below. Starting in the third line, explain the difference between average velocity and average rate of change of velocity with respect to clock time.
We can sometimes use the initial and final values of a quantity to determine the average value of that quantity. The rule is as follows:
Rule 2: If we know the value quantities A and B at the beginning and the end of an interval, and if we know that the rate of change of A with respect to B is constant on that interval, then the average value of A on that interval is equal to the average of its initial and final values.
Definition of average of two quantities: To get the average of two quantities, add them and divide by 2.
Given the information you used previously, if we want to find the average velocity using the present definition, then what would be the quantity A and what would be the quantity B?
What are the initial and final values of the quantity A?
What therefore is the average value of the quantity A?
How does this differ from the process of finding the average rate of change of velocity with respect to clock time, and what special condition must hold in order to apply this definition?
If a ball on a track is at position 30 cm at clock time t = 3.4 seconds, and at position 80 cm at clock time t = 8.9 seconds, then how do we use the above process to find the rate of change of position with respect to clock time?
To apply the definition of rate of change you again follow these steps:
If you know that the rate of change of position with respect to clock time is constant, then how would you apply Rule 2 and what quantity would you get from Rule 2?
What is the significance of this quantity for the given situation?
For the given situation, what does it mean to say that the rate of change of position with respect to clock time is constant?
If you know that the velocity of an object changes from 20 m/s to 30 m/s over a time interval spanning 5 seconds, then what rule do you use to find the average rate of change of velocity with respect to clock time over this interval? State specifically how you use the rule, showing every step of the reasoning.
If in addition you know that the rate of change of velocity is constant, then what additional quantity can you determine? State specifically the rule you use and explain every step of the reasoning.
If you know the average velocity and the change in clock time, then for this situation what additional quantity can you find using Rule 1?
Find the value of this quantity for the given situation. Again be specific in your application of the rule, being sure to identify quantity A and quantity B, restating the rule in terms of these quantities and applying it in detail to the question.
If you know that the average rate of change of the velocity of an object, with respect to clock time, is 10 cm/s^2, and if you know that the object's velocity at t = 4 seconds is 20 cm/s, then how do you apply Rule 1 to find the object's velocity at clock time t = 7 seconds?
Again identify the quantity A and the quantity B, rewrite the rule in terms of these quantities, substitute the information you are given and explain every detail of your reasoning.
If you know that the average rate of change of the position of an object, with respect to clock time, is 20 cm/s, and if you know that the object's position at t = 4 seconds is 180 cm, then how do you apply Rule 1 to find the object's position at clock time t = 9 seconds?
If in the preceding example you also know that the object was initially moving at 10 cm/s, then what additional quantity can you find by applying Rule 2? Again be very specific in your reasoning and your details.
What additional quantity can you now find applying Rule 1?
Before continuing to graphical interpretations, you should refer to the Trapezoids document and you should be sure you understand everything presented there. You should also review the information on trapezoids presented in the q_a_initial_probs program under Areas (you did this exercise as part of your Orientation so this document along with your responses and instructor commentary should be posted at your access site)
Graphical Interpretation
On a graph of A vs. B, the quantity A is represented relative to the vertical axis and the quantity B relative to the horizontal. Between two graph points, therefore, the change in A is the 'rise' from one point to the other, and the change in B is the 'run' from the same first point to the second. The average rate of change of A with respect to B is then
where for brevity 'ave rate' is understood to mean 'average rate of change of A with respecct to B'.
Since rise / run between two points is the slope of the straight line segment between those points, we can identify an average rate of change of A with respect to B as the slope between two points on the graph of A vs. B.
Since slope = rise / run, we see that
Interpreting slope as ave rate, rise as change in A and run as change in B, this again tells us that
Note that the data program is in a continual state of revision and should be downloaded with every lab.
Your instructor is trying to gauge the typical time spent by students on these experiments. Please answer the following question as accurately as you can, understanding that your answer will be used only for the stated purpose and has no bearing on your grades:
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... examine ave roc of position with respect to vel, or vel wrt pos