differencequotient_lab

course Phy121

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

I submitted the form once again without my contact name and information included. I submitted it at approximately 11:05 a.m. on 2/27. I am including it here as well so you can compare. Sorry about that. " "Form ConfirmationThank you for submitting the following infor identifyingInfo: Submitting Assignment: Deterioration of Difference Quotients yourCourse: Contact_FirstName: Contact_LastName: email: Your optional message or comment: Your report of the two graph coordinates that most clearly lie above the best-fit curve: 40,36.972 80,4.96 Your report of the two graph coordinates that most clearly lie below the best-fit curve: 30,49.178 60,15.453 Is the straight line more or less steep than the curve between t = 30 and t = 40? The straight line would be less steep than the curve, but only slightly. Since the point at 30 is below the curve, and the point at 40 above, that will tilt the line to a smaller slope. It will be less steep. Is the line segment steeper or less steep than the curve between t = 70 and t = 80, and between t = 80 and t = 90, and are the answers related? Between 70 and 80 I think the curve is less steep. The change in acceleration is slowly decreasing as time increases. There is a steep drop at the beginning of this graph, then it slowly starts levelling off towards the bottom. So overall, between 70 and 80, it's not as steep as the steepness of the overall curve. Between 80 and 90 it is less steep as well for the same reasons. It is levelling. Yes, as the graph nears the end of the time segment it is coming to a straight line, as opposed to the beginning when it was dropping dramatically in its change in position. Difference quotient results for original data: -1.701 -1.547 -1.221 -1.115 -1.037 -.6063 -.443 -.4473 -.042 Your explanation of the calculation of the first difference quotient, its graphical meaning, and its meaning in terms of the object being observed. The difference quotient was calculated by taking the change in the 2nd column quantity (in this case, the change in position) and dividing it by the change in the 1st column quantity (or time). Since the 2nd column quantity is the object's change in position on our graph, this represents the rise of the graph, and 1st column represents time, or the run. So this difference quotient tells us the slope of the line segment between two points on a graph. Since this is the change in position/change in time, this also represents average velocity for the object at that point. Your explanation of why the difference quotients are negative, what this has to do with the graph and what this tells us about the behavior of the object being observed The difference quotients are negative because the object is decreasing in its velocity over the time period studied. As it decreases its velocity, it is decelerating so its change of position is also slowing down as time continues. Thus, the line in the graph is decreasing at a decreasing rate. The object is decelerating and ""covering less ground"" as it decelerates. The object is slowing down and coming to a stop. Are the difference quotients increasing or decreasing, and do they appear to be doing so at an increasing, decreasing or constant rate? The difference quotients appear to be decreasing, as we are approaching zero and the object is slowing down. I could not tell at what rate however, it seemed to fluctuate and was not constant. The differences varied. Is this what is happening? Granted, the differences are not great, so accounting for errors, it could possibly be decreasing at a constant rate. Your explanation of why the ave vel vs. midpt clock time table might be said to give us the most accurate velocity vs. clock time approximation we can obtain directly from our data points. The table provides us with the average at the midpoint since there can be some errors in our observation of the points on the graph. The table takes the original data we entered and can provide us with average velocities based on this data. Your explanation of how the difference quotient vs. midpoint clock time graph shows the trend of the velocity of the system, and also shows the fluctuations resulting from the original uncertainties: Since the graph has an upward slope representing the velocity of the object, we can see that over time the object is slowing down, decelerating and eventually probably going to stop. This shows the fluctuations from the original uncertanties in observation since the points seem to veer off the course so to speak. If an object is rolling down an incline at a constant acceleration/deceleration, the change in velocity should be close to constant over time. In this graph we show some large gaps in between points (around 35 and 65). As I had stated above when asked if it was decreasing at constant rate, based on the actual points it was difficult to say since the difference among the points varied. How do you think these straight lines differ from the actual behavior of the system? I think the straight lines here do not represent an accurate picture of the behavior of the system. There isn't a sudden sharp decrease in the velocity of the object at a specific time, because there would be no way we could observe that in real-time. If a object is rolling down an incline it will slowly come to a stop, and rarely will make a sudden decrease as it decelerates. A straight line is a better picture form the decrease in velocity over time. How does the best-fit line better represent the system than the broken-line graph? As stated above, an object rolling down an incline will do so at a constant change in velocity, without any sharp fluctuations in speed. A straight line represents this constant change as opposed to the points that vary across the time period. Is the best-fit velocity vs. t line completely accurate or is there some uncertainty? I would expect random uncertainties still remain. Unless this is a ""perfect"" system, there would be no way to account for all factors that could be affecting the velocity. I think the straight line, however, represents the closest we can possibly come without involving elaborate mechanisms to achieve the ""ideal"" situation. Your copy of difference quotient vs. midpoint clock time, for v vs. t results. 20,-.0154 30,-.0326 40,-.0106 50,-0.007800 60,-.04307 70,-.01633 80, 0.0004299 90,-.04053 Why does this last table show only 20 to 90 seconds? The data we entered here started at 15 and ended at 95, so we are now looking at the midpoints between those 2 points, so we would start at 20 and end at 90. Do you see a clear trend in the new difference quotients? The differences between are somewhat constant? I don't really see a clear trend. If there is one, I am missing it. It tends to go up and down, decrease/increase/decrease/increase. Your explanation of the calculation of the first of the new difference quotients, its graphical meaning, and its meaning in terms of the object being observed. The difference quotient was calculated by taking the change in the 2nd column quantity (in this case, the average velocity) and dividing it by the change in the 1st column quantity (the midpoint of the time interval). Since we are taking the difference of the average velocities over the midpoint, and we're using the same data, they will probably be more related to the broken-line graph. Once again, the object's velocity is decreasing since we have a negative value. Do these new difference quotients represent the slopes of the broken-line graph or the straight-line approximation of the best-fit v vs. t line? If these represent the slopes of the lines, I am not sure they represent either line, they fluctuate up and down. Is there a clear trend to the acceleration? No, I do not know how I would draw this line as they increase and decrease throughout. No, I cannot see a clear trend, it is very difficult to get a sense of what it is doing. Your understanding of how repeated application of the difference quotient can magnify uncertainties. Using the original data, which contained errors and uncertainties, and using it repeatedly in more exercises, calculations, etc. seems to magnify these errors. I think since we are using the difference quotient, and ""narrowing"" down our data, as we do this, it brings out the errors clearly. I definitely see the difference from our original graph versus the one with the midpoints. I cannot tell what the trends are. Describe how slight jaggedness in the position graph becomes significant jaggedness in the velocity graph and extreme jaggedness in the acceleration graph. Yes. With each calculation we seem to come closer to the raw data points, and as we do, we clearly see how inconsistent and erroneous they can be. It appears to me to be a matter of scale? The more accurate we need to be, the more we can clearly see our errors. I imagine if we can many data points, hundreds or thousands, we would see less of this error? Would a lack of a trend in our acceleration graph indicate that there is no actual trend in acceleration, or would it be the result of uncertanties or our choice of method of analysis? I think there is a trend in the acceleration of the object, and that our inability to observe it accurately has affected this graph, and possibly our difference quotient. For this type of analysis did we have enough data to run it using this method? Is there a more appropriate analysis available for this type of data? Were there steps we could have taken in our observations to help decrease this type of error? Is there a way to account for these errors in the analysis method we use? It's hard to say to what extent our observations vs. the method of analysis played as I am not all that familiar with the difference quotient. Your description of the difference in the quality of information as a result of reducing experimental uncertainty by a factor of 3. Based on the last graphs above, we can clearly see a greater difference from our original graphs. The intial step of analysing with the difference quotient shows a much better representation of constant acceleration than our original. The points are closer to the curve. It is especially prominent in the difference quotient vs. midpoint clock time, representing approximate velocity vs. clock time. Here there is a remarkable difference from our original line. The data look cleaner. Your comment on the potential usefulness of mathematical smoothing vs. repeated application of difference quotients. Optional additional comments and/or questions: Return to the form. "

I wondered whose work that was. Thanks for clearing it up.
I have to process your work in order to post it properly, and I need you to go back to the form and fill in the information up through your email address. Once I get that line I'll be able to put it together with your information so the filing program will be able to work on it.
In any case the work looks good. Let me know if you have questions.