Total marks available: 24

Total marks available: 24

Assessment contribution to overall grade: 10%

Instructions for the Practical Activity #1 worksheet

Complete all tables and prompts for answers (identified by grey underlined text).

Results of any calculations should only be rounded once as the final step when you quote your result. Keep enough precision during the intermediate steps to ensure you have appropriate precision in your quoted result.

Marks for explanations are awarded for clearly drawing on the correct ideas and logically linking these ideas together to interpret your results.

You can type in your answer or can use the Wacom tablet to ink-in your answer. Feel free to adjust any spacing to fit your written text in.

Describe the experimental setup you are using and how you are conducting the experiment and making your measurements of fall distance. A clear picture illustrating your setup should be included.

I am using a ruler for this experiment, with my helper holding the ruler at the 0cm mark for as accurate as possible measurement of distance of fall when I react and catch the ruler.

Click the box below to embed the picture of your experimental setup.

Table 1. Measurements and analysis of fall distances to determine fall times.

Attempt y

(Units) Error in y

(Units) % Error

in y t(Units) % Error in tError in t (Units) Quoted result for t (Units)

1 2 3 4 5 6 7 8 9 10 Explain your reasoning for either including, or not including, any additional error on the fall distance, y.

Click here to enter your response & reasoning.Are there any outliers in your dataset? Clearly identify which attempt(s) you believe are outliers and explain your reasoning for identifying these as outliers.

Click here to enter your response & reasoning.Determine the mean of your reaction time.

My reaction time is Click here to enter your mean.

Table 2. Statistical analysis of reaction times

Measurement Mean Deviation from mean Square Deviation

Sum of square deviations: Standard deviation: Standard error: Enter your quoted reaction time, including standard error:

My reaction time is Click here to enter quoted reaction time.Which do you think had a greater impact on your results in this experiment: Errors that arose from individual measurements, or errors that arose from the natural variation of your reaction time from attempt to attempt? Explain your reasoning.

Click here to enter your response & reasoning.Reaction time used from forum = Click here to enter reaction time from forum.Can you say whether your reaction time is the same or different to the reaction time taken from the forum? Explain your reasoning.

Click here to enter your response & reasoning.

In this activity you will be conducting an experiment to determine your reaction time. The learning objectives for this activity are to:

Conduct a simple experiment and record experimental data.

Analyse experimental data.

Include errors when recording and analysing experimental data.

Interpret your analysis to determine whether experimental results agree or disagree.

Appreciate human limitations when recording experimental data.

Theory

Reaction Time

You cannot respond instantly to events happening around you for example if you touch something hot, some small amount of time is required for the nerve cells in your hand to send an electrical signal to your brain, and then your brain to send another electrical signal to the muscles necessary to move your hand away from the hot object. This time interval between the stimulus or event (you touch the hot object), and the action (you move your hand) is called your reaction time.

Your reaction time is one example of a human error a type of uncertainty that may be present in data collected when conducting an experiment. People have a range of reaction times, typically in the range of 150 300 ms.

Free Fall

When we talk about acceleration, we mean the amount by which the velocity of an object changes per unit time. We can write this as:

Acceleration=Change in velocityTime interval=vtIf the change in velocity is in the standard units of metres per second (m/s) and the time interval is in seconds (s), then the acceleration is in units of metres per second per second, or metres per second squared (m/s2). In many situations, such as when travelling in a car, the acceleration constantly changes as the velocity changes (which can be caused by either the speed and/or the direction changing).

For an object moving under only the force of gravity, i.e. no other forces acting and any air resistance can be ignored, the acceleration of the object is constant and given by the acceleration due to gravity, which we denote with the symbol g. This type of motion is called free-fall.

The equations of motion allow us to calculate the properties for the motion of an object that has a constant acceleration. An object in free-fall that is released from rest, and has moved under the acceleration due to gravity for a time interval t, will have fallen a distance y given by:

y=12gt2( SEQ Equation * ARABIC 1)

Which can be re-arranged to give the time interval as:

t=2yg( SEQ Equation * ARABIC 2)

Experiment

In this experiment, you will be trying to catch an object released from rest. As the distance the object falls before you grab it (y) can be measured, you can experimentally determine your reaction time (t) using equation (2), knowing that the acceleration due to gravity at the surface of the earth is g = 9.80 m/s2.

The falling object could be a ruler (if long enough) but could also be something long that you can easily grip, for example a broom or mop handle. You will also need an assistant to participate in this experiment your assistant will hold the object to be dropped, and you will grab it when you notice it is falling. You cannot validly conduct this experiment on your own.

If you are not using a ruler, you will need to mark a point on the object as a reference position so that each time the object is dropped it can be positioned and released consistently, and the distance fallen can be reliably measured. If using a ruler, take the 0 cm mark to be the reference position. Your assistant holds the top of the object, letting it hang vertically with the reference position at the free end. You should position your thumb and index finger held open with the free end of the object, close to but not touching the reference position, so that you are ready to grab the falling object once you notice it is falling.

Your assistant will release the object without saying or doing anything that gives you any indication that they will release it. Once you notice the object is falling, you should close your thumb and index finger to grip the falling object. The distance between the reference point and where you have gripped the object is the fall distance, y. As you work through the data collection and analysis you may wish to consult Appendix 1: Errors and error analysis in experiments, which is at the end of this Theory and Instructions file.

Document the experimental setup you are going to use on the worksheet. Ensure you include a picture of your setup/equipment used to illustrate how you are determining the distance.

Attempt the object-drop experiment up to 10 times, measuring the fall distance from the reference point to the grab point. Repeat the drop as many times as you feel is needed to be confident in the consistency of your results. Enter the result for all of your drops, including any bad drops, into Table 1 on your worksheet.

For each measurement ensure that you record the error in your measurement of distance. The error will be at least as large as the least count of measurement device but could be larger if you think there are additional errors involved based on how you conduct the experiment.

Explain your reasoning for either including, or not including any additional error on the fall distance y.

For each fall distance, y, determine the time interval, t, that the object was falling for using equation (2). Enter your results into Table 1.

Determine the percentage errors for each of your measurements of fall distance. Enter these into Table 1.

For each attempt, use your percentage error in the fall distance, y, to determine the percentage error in the fall/reaction time, t. Record these in Table 1 also.

Using your calculated values of, and percentage errors in, t, determine the errors in t. Record these in Table 1.

Finally, complete the Quoted result for t column by appropriately quoting your result for each attempt in the form, ResultError.

You now have a number of measurements of your reaction time, t, with errors. When making multiple measurements or calculations of the same quantity, it is natural to get a little variation in that quantity. But sometimes you obtain results which clearly do not agree these results which are clearly not in agreement are outliers. These outliers should be kept in the recorded results, but not included in any further analysis.

Are there any outliers in your dataset, i.e. quoted reaction times which do not overlap with most of the rest of the results? Clearly identify which attempt(s) you believe are outliers and explain your reasoning for identifying these as outliers.

Determine the mean of your reaction times, neglecting any outliers in your calculation.

Using the mean result for your reaction time, the information around random error in Appendix 1, and Table 2, determine the standard error in your reaction time.

Quote your mean reaction time with the standard error.

Which do you think had a greater impact on your results in this experiment: Errors that arose from individual measurements, or errors that arose from the natural variation of your reaction time from attempt to attempt? Explain your reasoning.

Post your quoted reaction time in the Experiment 1 Discussion post on the forums. Compare your quoted reaction time to someone elses quoted reaction time. Can you say whether your reaction times are the same or different? Explain your reasoning.

Appendix 1: Errors and error analysis in experiments

Every quantitative measurement in science raises questions of accuracy and precision. For a single measurement, accuracy is how close the result is to the expected or actual underlying value, whereas precision is how precisely the result is known, i.e. how many digits for the result can we reliably give. All measurements also require an error which indicates that the quoted result might be some amount larger or smaller than the measurement or calculated result. Errors are usually indicated using a symbol, with the amount of error to the right of the symbol: Result=Measured valueerror. The amount of error limits how precise a measurement can be made.

Suppose, for instance, that the length of a swimming pool at an Olympic Games is measured and a value of 51.3 0.2 metres is obtained. This result exhibits a high degree of precision because the error is very small compared to the measured value. However, since the pool is known to be exactly 50 metres long (give or take a few millimetres), the measurement is not accurate because the expected value of the length (50 m) lies outside the range of the measured value with its error (i.e. 50 m is not between 51.1m and 51.5m).

That same length could also be measured by someone else using different equipment and/or measurement techniques and obtain a result of 49 2 m. This result has a low degree of precision as the error is relatively large compared to the measured value. However, the result is considered accurate because the expected value (50 m) lies within the range of the measured value with its error (i.e. 50 m is between 47 m and 51 m).

Experimental errors

All experiments, however, carefully designed and performed, involve errors due to limitations or faults of the apparatus or the experimenter, and statistical fluctuations due to environmental factors, or the equipment & experimental techniques used. Such errors should be considered, and an estimate of the probable errors included in any discussion of results. Usually an experiment includes one or more measurements which limit the precision of the final result. In this case the experimenters effort should go into minimising errors in these measurements. It may even be unnecessary to measure other quantities to the maximum possible precision. For example, if one measurement cannot be made to lower than 10% it may be a waste of effort measuring other equally important quantities to 0.1%.

Uncertainty in the final result may arise from two types of error: systematic errors and random errors.

Systematic errors

Systematic errors may be introduced by the experimental conditions, e.g., temperature variations, equipment precision, or through faulty or mis-calibrated equipment. The very presence of the measuring instrument may change the phenomenon being investigated. Such errors are consistent throughout the experiment and hence cannot be diminished by any statistical averaging process. A well-designed experiment can often reduce or even remove such errors. Where this is not possible, the errors should be investigated, and the necessary corrections made in the quantities being measured. It is not possible to give detailed advice as to how systematic errors may be overcome. Each experiment must be considered individually and only by a thorough understanding of the objective of the experiment and the techniques being used is this possible.

An example of systematic errors are instrument measurement errors. Each measurement instrument is only able to measure to a certain precision. The instrument measurement error for a particular measurement can be estimated by determining the least count of the measurement device, that is the smallest quantity, or difference in that quantity, that the measurement device can sense. For example, a mass balance that can display the measured mass to the nearest gram would have an error of 1 g, or on a ruler which could have an instrument error of 1 cm, 1 mm, or 1 inch, depending on which measurement scale is used.

Sometimes there are additional errors present in the experimental technique that cause the error in the measurement to be larger than the least count. You may need to estimate the size of these errors yourself based on the equipment used, the experimental techniques and methods used, or the methods used to make the measurement.

Random errors

Random errors occur in all measurements. Suppose that you, taking all possible precautions against known errors, were to make a measurement of the same quantity ten times. You would expect to obtain results which differ slightly each time the measurement is made.

As an example: you make a set of three measurements which are: 191, 202, 207.

Assuming that each measurement was of the same quantity and obtained using an identical method, the most accurate value is the arithmetic mean (the common average). In this case, the arithmetic mean (average) is [(191+202+207)/3] = 200 (obtained by adding the values and dividing by the number of measurements).

Quoting the arithmetic mean by itself is insufficient, however, since the mean value is not necessarily the true value. We need an estimate of the amount by which the mean may be in error. There are a variety of ways of making this statement about precision. Some accepted ways are:

the rangea simple statement of the difference between the largest and smallest of a set of readings (the range here for this example is 190 210),

the standard deviation or root mean square deviationa quantity found by calculating the average of the squares of the deviations from the mean and taking the square root of the result.

The deviation of each measurement from the average (200) is shown in the next table:

Measurements 191 202 207

Deviation from mean 9 2 7

Square Deviation 81 4 49

Each number in the Deviation from mean row is obtained by subtracting the mean (200) from the measured value. The square deviation is obtained by squaring the deviation from the mean.

The standard deviation () is computed by adding up the square deviations ((SquareDeviations) = 81 + 4 + 49 = 134), then dividing by the number of measurements (134/3=44.6) and then by taking the square root (44.6=6.68).

Statistical theory shows that the best estimate for the error in the mean is the standard error. The standard error is obtained by dividing the standard deviation by the square root of the number of readings. In the above example, the error in the mean of the 3 readings is 6.68/3=3.85. We always round the standard error up, so the standard error here would be quoted as 4. The result of our analysis here would give the result as: 200 4.

Error Propagation

The amount of error in a measurement will impact the amount of error from the result of a calculation which uses that measurement. Following the errors through a calculation is an called error propagation.

Say we are conducting an experiment to determine the acceleration due to gravity (g). We could measure the time (t) an object takes to fall a measured known distance (y) when released from rest, and then calculate g using:

g=2yt2( SEQ Equation * ARABIC 3)

The measurements of time and distance each have errors that need to be followed through this calculation. A simple method involves combining the percentage errors in the measurements to determine the percentage error in the calculation result.

The percentage error can be calculated by dividing the error by the measured value, then multiplying by 100 to obtain the percentage error:

%error=errormeasurement100%( SEQ Equation * ARABIC 4)

If the measurements were: t=0.2940.002 s, and y=4195 mm, then the percentage errors in each would be as summarised in the table below.

QUANTITY MEASUREMENT ERROR % ERROR

y419 mm 5 mm 1.19 %

t294 ms2 ms0.68 %

t2- - =20.68 %=1.36%Another result from statistical theory is that whenever quantities are multiplied or divided, the percentage errors add. t2 is the same as t t, so the error in t2 would be twice the error in t, which is calculated and shown in the table above. The same applies for any power, for example when taking the square root, we can write this as to the power of one-half: 5=512, so the error in the square root of a quantity would be half the error in that measured quantity.

Because we are multiplying and dividing for this calculation, the percentage error in g would be the sum of the percentage errors, i.e. 1.19 % + 1.36 % = 2.55%. A calculation of g using the measured values yields:

g=20.419 m0.294 s2=9.695ms2( SEQ Equation * ARABIC 5)

Which for a percentage error of 2.55% would yield an error of 9.695 m/s2 0.0255 = 0.247m/s2. The calculated result for the acceleration due to gravity would therefore be: 9.6950.247 m/s2.

Quoted results and discussion

The final step is to apply the rounding rules to appropriately quote our calculated result:

First round the error up to 1 significant figure (0.3 m/s2).

Then round the calculated value to the same precision as the error (9.7 m/s2). By the same precision it means either to the same number of decimal places if the error is smaller than one, or the same number of 10s or 100s or 1000s if the error is larger than one.

This gives the appropriately quoted calculated result as: g=9.70.3 m/s2. The result of g from this experiment could be as low as 9.4 m/s2, or as high as 10.0 m/s2.

By comparing the accepted value of g=9.80 m/s2 to the experimental result g=9.70.3 m/s2, the accepted value falls within the range of the error, so the measurement would be considered accurate.

Suppose the experiment were conducted again, and a result of g=10.070.02 m/s2 was obtained. The maximum and minimum values of g from this second result would be 10.09 m/s2 and 10.05 m/s2. The ranges from the two experimental results do not overlap, so the results do not agree.

The above concepts and calculations represent an exemplar for you to understand and perform calculations such as mean value, standard deviation, percentage error, standard error, and error propagation in experiments that you conduct.