The Efficiency of a Super Ball

In this lab our purpose was to determine the efficiency of a superball as it is dropped from different heights . Efficiency is the ratio of energy out to energy in. When a ball strikes the ground, it does so with a certain amount of kinetic energy due to its speed. During the bounce, the ball deforms as the kinetic energy is converted into elastic energy. As the ball rebounds, the elastic energy is returned to kinetic energy. This process is not perfect however and heat energy is generated as molecules are pushed against each other. Also energy is lost to the floor and to the air as vibrations and sound. If the ball rebounds with 90% of the kinetic energy that it had before the impact then we say it has an efficiency of 90%. It is possible that the ball’s efficiency is not constant; that when it is dropped from a high height, landing with great speeds, it is more or less efficient than when at lower energies. Our answer, therefore may not be a single number, but instead it might be a function of how efficiency varies over drop heights.

We make several assumptions in this lab: we assume that air resistance is negligible and therefore that the entire amount of potential energy (mgh) that the ball has due to its drop height is

converted into kinetic energy ( m . ( m and )

We also assume that the all kinetic energy after the bounce returns to potential energy as the ball slows and reaches its new bounce height.

To answer the question posed in this lab we will compare the heights from which we drop the ball, to the heights to which it rebounds. Since the kinetic energy that the ball has immediately before and after the impact are equal to the potential energy that the ball has at its high points before and after the drop (due to our assumption of no significant air resistance) and since the height of the ball is directy proportional to its potential energy, we can determine the ball’s efficiency by the ratio of final height /starting height.

Efficiency =

By graphing the efficiency vs. the starting height we will be able to see how this efficiency depends on height. We will also show how

Purpose is established

Key terms defined(efficiency, etc,) in the context with which they will be used. Not stand alone definitions.

Relevant background information is provided to help make sense of ideas and future conclusion.

Overview of method and how question will be answered.

Frequently you will use a drawing to help make your ideas more clear.

You can choose your own title as long as it’s relevant.

the efficiency depends on impact velocity by graphing it against velocity too.

Methods

Data and Calculations

Raw Data

Starting Height (cm)

Final Heights(cm)

Trial 1 Trial 2 Trial 3 Average40 32 35 33 33.350 41 41 41 4160 48 48 46 47.380 60 64 65 63100 80 79 81 80120 100 96 98 98150 117 123 120 120200 160 160 162 160.6

Calculated Values

Starting Height (cm)

Efficiency

40 .8350 .82

Final Height

hfinal

We dropped a superball three times from each height listed on the data table. Each time we measured the starting height and the highest rebound point, based on the bottom of the ball. We used a meterstick placed vertically on the floor for our measurements.

Starting height

hstarting

Clearly defined sections to lab report.

Key variables clearly shown on a diagram.

Nothing fancy, but if it were complicated it might take two diagrams or an explanation of how to get a measurement.

Efficiency =No need to show sample calculations for simple operations or averaging.

Show relevant formulas.

Units.

Clear labels on columns.

All data shown. Sometimes a number may be so odd that you don’t want to include it in your calculated values, but you should still include it in your tables.

Frequently the data tables on logger pro get cut off. Make sure your data tables are complete.

Even if you’re referencing someone else’s graphs, you need data tables.

60 .7980 .79100 .8120 .82150 .8200 .8

Velocity before impact (m/s)

Efficiency

2.8 .833.2 .823.5 .794.0 .794.5 .84.9 .825.5 .86.4 .8

ConclusionThe data from this experiment shows a constant ratio of height-out to height-in, meaning a constant efficiency for a super ball. We also see this result in both of the next two graphs showing a constant efficiency of about 80%, whether graphed against drop height or

V=

Sample calculation

At 40cm

V= =2.8m/s

Feel free to include hand written calculations.

I’m not including graphs here because this is a simple word document and I encourage you to use Logger Pro for your graphs.

Graphs should have well labeled axes with units.

Whenever possible data points should have error bars showing your uncertainty about the data points.

Dot to dot lines are never appropriate.

Instead use trend lines (sometimes curves). Include equations for these lines.

Summarize the data.

Address/answer the question explicitly.

In this case, the data supported the conclusion clearly and in a straightforward way. It would have sounded silly to belabor it, but sometimes it’s complicated or ambiguous and takes a full paragraph.

impact speed. The slope of the first graph 0.807 indicates the efficiency to three digits of accuracy, but as I will describe in the next section, this surely overstates our accuracy. Apparently no matter the amount of compression of the ball during impact, the intermolecular dampening and heat generation causes the ball to lose about 20% of its energy.

Limitations and Error AnalysisDespite this apparently very clear result, there are several reasons to doubt this result.

First, we made the assumption that no energy was lost to air resistance. During the bounce the ball may be more efficient than 80% and some of this loss may be due to air resistance. However, upon reflection, I think ignoring air resistance remains a good assumption. If air resistance was significant then the higher drops should have experienced a greater percentage of energy loss. That they didn’t, indicates that air resistance wasn’t important and the bounce efficiency really was 80%. (Or that there was a big coincidence about factors exactly canceling out.)

More importantly, we only tested a limited range of heights. We know that eventually balls stop bouncing, meaning that from some tiny drop height nothing happens, 0% efficiency. This probably happens when the compression between molecules during the bounce is on the same order of magnitude as the random motion of the molecules due to their temperature. We also know that given enough speed a superball will break as the compressional forces exceed the intermolecular forces. To improve this experiment it would be good to expand the range of data collection to investigate this nonlinear behavior and each end of the data.

Finally, while 80% efficiency is probably a reasonably accurate result, numerous measurement limitations prevented us from getting a more accurate answer. We estimate that we could hold the ball at our established drop height within 0.2cm of accuracy. Our bounce height measurements were not nearly as good unfortunately. Sometimes our three watchers would disagree on a bounce height by up to 3 cm, and our multiple trials for one case spanned 6 cm (150 cm drop height). We indicated this uncertainty with 3 cm error bars on our final height data. Since 3cm represents a 3% uncertainty in our median drop height of 100cm, we decided that our final

When appropriate, propose a theoretical explanation for your results.

Describe your confidence in a general way. I do this a bit with the first sentence, and a bit more with the last.

Support your claim of confidence, or lack thereof, by discussing the strengths and weaknesses of your experiment. A range of possible approaches is given on the assignment sheet.

At least two aspects of your error analysis should have a quantitative element. Here I’ve described my original uncertainty quantitatively, the size of my error bars quantitatively and how the original uncertainty affected the calculated efficiency value.

efficiency values had a 3% uncertainty and put error bars of 0.03 on them. This probably overrepresents the uncertainty of the large drops and underrepresents the uncertainty of the small drops, but our graphing program has limited functionality for setting error bar size. It also assumes that the large uncertainty in the bounce height makes irrelevant the small uncertainty in the drop height.

Our actual calculated efficiencies are all within 3% of 81%, as seen in the fact that the line of best fit passes through the error bars of all of the points. This and the fact that the graph has a y intercept of 0, predicting that 0 drop height would not bounce, add to our confidence in our result of 81±3% for the efficiency of our superball no matter its drop height.

Error propagation (how original errors affect calculated values) can be a very complicated thing. I’m alright with your making some simplifying generalities like I have here.

Your error analysis part is where you can really show off a deep understanding of what’s relevant in the lab. Put some thought into it. Show off.