A Scientific Model for Free Fall.

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1 A Scientific Model for Free Fall. I. Overview. This lab explores the framework of the scientific method. The phenomenon studied is the free fall of an object released from rest at a height H from the ground. The steps of the lab are as follows: 1- Gather experimental data (height of fall H and time to fall T, for various heights) for the falling object. 2- Construct a model describing the phenomenon: you will use your data to propose an equation which relates the two variables H and T (either H = at+b, or H = at 2 b. ) This equation will constitute your model. 3- You will then test your model by predicting the time of fall for a different H and check the prediction of your model against experimental evidence. data. In this lab you use concepts in error analysis (see Web Material) as well as the use of Excel to analyze your II Quoting Measurements. II. A Error Analysis Please refer pdf file "Error Analysis: A Basic Introduction" available on the course website in the "Course Material" tab. II. B Significant Figures : Recall that the number of significant figures (sig figs) is equal to the number of figure in a number, counting from right to left until you reach the last non-zero digit. eg has 3 sig figs: 1, 0 and 2 and 200 also has 3 sig figs: 2, 0 and 0. The zeros to the right of a number are the reason why you must use scientific notation 2*10 2 has 1 sig fig but 200 has three, so does 2.00*10 2. In 200 the zeros are there to tell us that it is not 199 nor 201, but 200; wheras 2*10 2 is consistent with any number betwee 150 and 249. And 2.0*10 2 is consistent with any number between 195 and 204. Standard deviations can be quoted with 1 or 2 sig figs, more and it becomes ridiculous!

2 2 free fall 1.nb II. C How Experimental Results Must Be Quoted in Experimental Physics: Experimental results are quoted as Q=Q ± Q with Q given by the standard deviation quoted with 1 or 2 sig figs. Since Q gives the uncertainty in Q, it is clear that the number of sig figs in Q should be consistent with Q. For instance if Q= and we wish to use 1 sig fig in the standard deviation, then we first round off Q to Q=0.03. Then, assuming that we found Q = , Q=0.03 tells us that the second digit after the decimal point is uncertain in Q and thus the 4 is uncertain. It is clear that all the digits to the right of the 4 in Q are therefore meaningless. So we must now round off Q to its uncertain digit: we get 1.25 and thus quote the result as Q=1.25±0.03. In case you want to quote the standard deviation with 2 sig figs, say Q=0.029 then we keep the corresponding digits in Q. We thus have the result Q=1.248± It is important to respect these rules. There are few things that look more idiotic than a result with 15% uncertainty (comon in many of our labs) quoted with 9 significant figures because that is what the calculator display shows at the end of the calculation. II. D Propagation of Uncertainties There are rules concerning how uncertainties are propagated. For instance if Q=1.26±0.02 and R=0.05±0.01 how should we quote the product P=Q*R? The rules are simple and are derived quickly with some calculus - and we shall do it later. However the widespread use of software tools such as Microsoft's Excel has made it easy to perform large number of computations and avoid dealing with this technicality. In our example we would just compute Q i R i for each of the N values and compute the average and standard deviation of these N values of Q*R rather than use Q R and use the rule for getting (QR) from Q and R.

3 free fall 1.nb 3 III Free fall: the experiment. The experiment is straightforward: Take a meter stick and a stop watch as well as an object of your choice (make sure it is dense and compact; e.g. take a 100g weight). Drop the object from 4 different heights 1.5m; 2m; 2.5m; 3m. Drop the object 5 times from each height and measure the time it takes to fall to the ground. Record all the data. It is strongly suggested that the same person drop the object and time the drop in order to minimize delays and thus systematic errors. IV Analysis of the data and determination of the model In this analysis you will determine whether your data supports a model of the form H = at+b, or of the form H = at 2 b, (or neither!) for the relationship between H and T. Since you will be doing the analysis using Microsoft Excel this section is rather lengthy because it walks you through the use of Excel. However you will use the skills learned here throughout the semester and hopefully beyond. Open an Excel spreadsheet, and enter your raw data in table form. At the end of each column (or row) of values for a set of 5 trials, enter in the following cell the formula for the average of these values, and in the cell following the prvious one, a formula for the standard devation (N-1) of these same 5 values. NOTE! To enter a formula in a cell, click on the equal sign at the left of the formula bar (the white box above your spread sheet) and then type in the formula. If you need a function define by Excel (like average or stdev) click on the drop down menu to the left of the equal sign and click on the function you need to insert. NOTE! Excel will prove much faster to use than entering in numbers in your calculator if you use the following feature: Copying a formula to other cells. Assume you entered your 5 trials in column form (for instance for the first height, h=1.5m, the times are entered in cells C5 to C9). Now let's put our averages in, say, row 10. So in cell C10 we enter: =(C5+C6+C7+C8+C9) / 5. Now for the next height in column D, instead of entering in D10 the eqivallent formula (= (D5+D6+D7+D8+D9) / 5, we can instead just copy the formula in C10 and Paste it in D10. Excel will take care of changing the C's into D's for us. And we can also do this even more quickly by selecting C10 and dragging the bottom right corner handle across row 10 where we want to enter the averages. The copying is then done automatically! Now you can have a shot at finding an empirical model for free fall since you have a table of heights as a function of times. Let's be very optimistic and hope that the model we are seeking actually relates H and T through a linear relationship (the simplest model possible).

4 4 free fall 1.nb Thus you need to check whether your data supports a model of the form: H = at + b. In order to do so the best is to actually plot H vs. T for the 4 heights and the 5 average times, and check whether or not we can plot a straight line through these data points. Thus you should now insert a chart in your spreadsheet (type: x-y scatter with x data series=cells showing the average times and y data series=cells showing the heights). Chances are that the points do NOT line up exactly. But you cannot discard the linear model quite yet. Indeed from the section on error analysis above you know that these data points are NOT points since the measurements must have an error associated with them. So you need to add error bars to these data points. The uncertainty on the height measurement is small enough (assuming you didn't make a big systematic error!) that you don't have to worry about it (say 2mm in 1m or 2/10 %! Try to draw the error bar on your graph: it won't even show!). Thus, let's just worry about the error in time (the horizontal error bar, right?). You already know the uncertainty in the times since you have calculated their Σ. The problem is to indicate them on the chart: Click on the data series (your data points) in your chart, the chart tools should be available on the ribbon. Click the "layout" tab and you should see an "error bars" button on the right, click on it and MAKE SURE TO CHOOSE THE "more error bars options" item. When you do that you should see both horizontal and vertical error bars on the plot. At that point move the mouse to select the vertical error bars on the chart (they should already be selected) and delete them. Then select the horizontal ones and edit them by clicking on the buttom "custom" and "specify amount". Then specify the amount (both positive and negative) by selecting your set of standard deviations. Now, can you tell whether or not your data supports, within the limits of the incertainties, a linear relationship between H and T? The answer is no, not without actually drawing a linear relationship, i.e. a line, through the data points. To do so, you can either print the graph and manually draw a straight line through the data using your judgement to draw the best fit line or (this is easier) ask Excel to draw the best fit line through the data: Right click on the data, and click on Add Trend Line, then on Linea, then click on the Options Tab and check the box to write the equation on the chart. Now you have a line through your graph and the equation of that line. Most probably the line will not go through all points, or even any (!), but if it crosses all error bars you can conclude that your data is consistent (or even supports) a linear relationship within the uncertainty of the data set. NOTE! Whether you draw the line using your judgement or let Excel do it for you (using a least square fit), that line and thus its slope is the result of an arbitrary choice (yours or Microsoft's computer geeks). Excel's best fit line is probably "better" in the sense that it is justified scientifically by being a better statistical representation of the data: that line minimizes the square of the distances from the line to the data points. So what is your conclusion? Does your data support linear relationship model between H and T? If it does, what is "a" (the slope of the line)? What is b? You can compute "a" directly from the graph (but be careful not to use the data points since they do not necessarily fall on the line) or, and again this is easier, you can ask Excel to compute it for you: Click on the Trendline on your graph, you can now go on to the last part of the lab: the model prediction.

5 free fall 1.nb 5 Now you must try the next simplest relationship between which is a quadratic relationship between H and T: H = m T 2 + b. To do so compute the square of all the times. Just square any one of the cells with time and drag the resulting equation to square all remaining cells. Then compute averages and standard deviations. Graph the resulting data points (x T 2,y=H) on an Excel chart as you did above, including error bars. Draw a best fit line. Is the quadratic model supported by your data? V. Model Prediction Now is time to test your proposed model from the previous section Make a time of fall prediction from you chosen model for free fall and do this for a height significantly larger than the previous heights (say 4m or more). Again measure the times of fall experimentally and get an average and standard deviation for these. Do you have agreement? If you don't it invalidates your theory, and if you do it doesn't provethe correctness of your model but increases confidence in it. VI. Conclusion Make sure to address the following questions in the conclusion : 1-Which model, linear or quadratic, did fit better your data? (Explain). This is the model you chose to represent free fall. 2-What is roughly the x-intercept of your model? Can you give it a physical meaning? 3- How well did you prediction fare in the test? Can you explain discrepancies? If there is a discrepancy and a cause is provided ther MUST EXIST a LOGICAL connection between the two! (DO NOT RELY ON "HUMAN ERROR" EXPLANA- TIONS and make sure than a cause for the discrepancy actually justifies the discrepancy, rather than making it worse!)

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