Name:     Date:

Falling Bodies

In this experiment you will be asked to make repeated measurements of two different types of quantities, one a continuous quantity, the other counting the results of a random process. Each will be described in detail later in this handout. It is important to realize that errors are propagated in measurements and combined in a nonadditive manner. Uncertainties in measurements are discussed at length in the previous laboratory information, A-1 at an earlier class meeting.

OBJECTIVES:

  1. To realize that random processes influence the outcome of all measurements and that the error or uncertainty estimates are indications of these influences. In particular, you should be able to make measurements, report errors in these measurements (x +D x), report a calculated result based on measured quantities, and graphically display the net results.
  2. To become familiar with lab procedures and computer data analysis.
You might normally think that you should measure your reaction time with a complicated timing apparatus, and for certain events this might very well be appropriate. Here we will use a simple system that will give us a fairly accurate (close to a true measure) value for reaction time. We should be able to properly analyze the data, too, which is the real objective of the experiment. By using a simple set-up we should avoid complications due to equipment.

Later we will study a relationship between the time an object is falling and the distance it falls. Thus here we can use a simple metre (yes, metre is the correct spelling for the fundamental unit of length) stick to indirectly measure time. For now, we will use the relationship that
 
D = 1/2 gt2 (1)

where D = distance, t = time, and g = 9.8 m/sec2, the acceleration due to gravity in metres per second per second.  To be consistent in units, you need to use metres for distance, seconds for time and m/s2

APPARATUS: (per pair of students)

PROCEDURE:

  1. Each student will measure his/her own reaction time. In pairs, one student will hold the metre stick at the upper end. The second person (whose reaction time is being measured) places one hand (use same hand throughout) at the lower end of the stick at a position that can be reproduced repeatedly for about twenty measurements. The first person releases the metre stick without warning, and the second person closes his/her hand to grab the stick. It is advisable to rest the hand on a table so as not to move it vertically. Be careful not to watch the person dropping the metre stick, but rather the stick itself. Do not attempt to anticipate the drop, but watch for the stick to actually start motion. Your report will include results for each member of your team.
  2. Record the distance the stick fell each time. This should be done in tabular form with an adjacent column for recording times.
  3. Switch positions and repeat so that each member of the team has a record of data for his/her own reaction time.
  4. Perform data analysis.

Figure 1: Technique for determining reaction time.

DATA ANALYSIS:

We do the Data Analysis in two ways for the sake of demonstrating that the same result can be achieved by either of several methods. You may choose which method is most suitable for your equipment, calculator, graph paper, etc. It is not intended to restrict you in this lab. Rather, you are encouraged to develop creative ways to get at the objectives and analyze the results. Keep in mind that the data analysis section of your report is extremely important. Here is where you convince the reader of your report that you have collected valid, reliable information and that it leads you to a justifiable conclusion about the phenomena you are examining. We show the following two different methods of data analysis in very complete form to demonstrate a thorough analysis. Keep in mind that your individual analyses of labs will most likely be more simplified since you will use the computer for much of your analysis throughout the year. Be sure to include appropriate computer printouts as part of the data section.
  1. Make a table with columns representing:
    1. Distance, D
    2. Deviation of Di from D - This is the absolute value of the average distance less any individual measurement.
    3. Time, t
    4. Deviation of ti from t - As above, this is the absolute value of the average time less any specific time. Time is not measured directly. It is calculated from the distance.
  2. Calculate t and Dt. There are two ways this can be done. Samples follow.
    1. Method One: This is the way shown in your lab handout on measurement and analysis. Follow the example: Suppose I did the experiment only four times. (You are to do it twenty times.) The following distances are recorded:
      2.
      Observation Number Distance Metres Deviation Metres
      1 .55 .01
      2 .41 .13
      3 .62 .08
      4 .56 .02
      Average .54 .06
      N = 4; there are 4 trials
      D = .54; this is the average distance (2)
      a.d.= .06; this is the average deviation (3)
      A.D. = a.d./ ÖN = .06/2 = .03 (4)
      So Distance = .54 ± .03 metres (5)
      Following the recommended procedure for determining a function from its independent variable:
      D = 1/2 gt2 (1)
      Thus:
      t = Ö2D/g = Ö2(.54)/9.8 = .33
      Dt / DD » dt/dD (6)
      Dt » (dt/dD)DD (7)
      Where:
      dt/dD=d/dD (the derivative of time which is a function of distance, with respect to distance)
      =d/dD(Ö2D/g)=Ö2/g d/dD (D1/2)
      =Ö2/g (1/2 D-.5)=1/Ö2gD      		 (8)
      dt/dD = 1/Ö2gD = 1/Ö2(9.8)(.54) = .307 (9)
      Then:
      Dt = dt/dD DD = (.307)(.03) = .01 (10)
      So:
      t = .33 sec
      Dt = .01 sec 
      Thus we can report the average time and the uncertainty in time as:
      t = .33 ± .01 sec
    2. Method Two: In recording data the time for each measurement is calculated when measured. The table looks like:
      3. Observation Number Distance Metres Time sec Deviation sec
      1 .55 .34 .01
      2 .41 .29 .04
      3 .62 .36 .03
      4 .56 .34 .02
      Average .54 .33 .02
      a.d. = .02 (11)
      A.D. = .02/ sqrt4 = .01 (12)
Both methods yield the same results. The greater the number of measurements the better the results. In this case the results are more easily obtained by method one, so that the times do not have to be calculated for each trial. It could be that you have a graph or chart available to get the value of time from the distance, but hardly would one be available to give the result as precise as listed.

Since the measurements were made to determine one particular value, a graph of these results is not valid. Rather, the result of the measurements is a point on the graph. All the measurements were made to find the reaction time; i.e., the time from when you observed the metre stick starting to move until you caught it. This is one value, with some degree of uncertainty. You might find it interesting to compare the average values in the class and find an average value for the class. Would you expect the A.D. to be greater or less for the class than for any specific individual? These types of things should be included in your write-up.

You should consider and discuss how your data "looks," i.e., how consistent is it? Do you have most of the data clustered in one area? Do you have one or two data points that are inconsistent with the others? If so, why are they that way? You may be able to justify "throwing" them away. For example, suppose your partner anticipated the dropping ruler and one value is very small compared to the others. Or perhaps he or she was day dreaming and that value was significantly greater than the rest. If there is a good reason to "toss it away", explain this to the reader what and why you are doing so. Otherwise, you must keep these values. Do not just throw away values that are inconsistent. Indeed, sometimes such anomalies result in new discoveries when they accurately reflect how the world really works.

Now that you've learned the proper method, realize that making measurements and reporting data is an important part of any lab activity. With a large number of measurements dividing by the square root of N or square root of N-1 is not much different and in fact for useful significant fugures, yields essentially the same results. For this reason many people choose to use simply the standard deviation that can be readily calculated by Excel or other spreadsheet without going through the details of the above extra steps. You do need to make it clear to the reader just what makes sense. In any case you must report an average value and an uncertainty. If you explain to the reader what you are doing and have strong justification for such actions, there is no problem. Many people tell the reader up front that they choose to use the standard deviation for the uncertainty and why. It is as simple as that. For most purposes that is a reasonable estiumate of how much variance occurs in making certain measurements.

Notes:

  1. Be sure to put Limitations to the Experiment first in your write-up.
  2. Be aware of significant figures.
  3. You will use Excel to calculate the average value and uncertainty. Exlain to the reader what you are doing and why.
  4. Discussion must be clear enough so that the reader knows what you did and how good your results are. have fune.
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