Is there a quadratic (parabolic) relationship between the time and distance for a ball falling to the ground?

This laboratory explores the relationship between time and distance for an object moving at a constant acceleration. In this situation the velocity is changing.

- Determine the acceleration of gravity g
- Explore the nature of the mathematical relationship between time and distance for a falling object

Existing gravitational theory asserts that the distance an object falls when dropped is given by the mathematical equation:

$\text{distance}=\frac{1}{2}\text{gravity}\times {\text{time}}^{2}$

or

$d=\frac{1}{2}g{t}^{2}$

The theory predicts that graph of time versus distance should result in the half-curve of the start of a quadratic parabola as seen in graph 1.

This graph suggests that time and distance are not related linearly. That is, twice as much fall time results NOT in twice as much distance fallen, but in MORE THAN twice as much distance fallen.

Confirming the hypothesis that a time versus distance graph is a quadratic curve is difficult. We cannot determine the slope of a curve using a best fit straight line. The slope would be in centimeters per second (speed) but the slope is changing, the line is curved, which means the speed must of the falling object must be changing.

If the theory is correct and the relationship is a quadratic relationship ("x²"), then we can square the time values, divide by two, and graph the resulting values on the x-axis and the distance values on the y-axis. The result should be a straight line with a slope of g as seen in graph 2.

$\text{d}=\text{g}\left[\frac{{\text{t}}^{2}}{2}\right]$

This is just like y = mx except that for x we are going to graph half of the square of the time [t²/2]. If all goes well, this second graph should be close to a line. The values on your axes will differ from those seen here.

The units of slope for the second graph and of gravity in this laboratory are centimeters per second squared, also written cm/s².

Note that your graph based on your data from laboratory might not produce a line as smooth as that seen above. Small deviations from a smooth line are the result of small errors in measurement, not evidence that the theory is false. The whole pattern of the data would have to disagree with shape proposed to disconfirm the theory.

Laboratory teams will drop a ball timing the fall time for the ball.

Teams of three to four students will be formed composed of the following roles to facilitate measurements:

- Ball dropper and timer
- Meter stick holders (2)
- Recorder

- Start by dropping and timing the ball fall time from 100 centimeters above the ground.
- Move up by 100 centimeters and drop the ball from 200 cm.
- Move up by another 100 centimeters and drop the ball from 300 cm.
- Move outside up to the second floor and drop the ball from 400 cm.
- Drop the ball from 500 cm.

Notes

- The team member who drops the ball also must be the timer. Only the person dropping can start the stopwatch precisely enough to get accurate time measurements. Practice first using the stopwatch. The theory and known acceleration of gravity suggest a time of around 0.45 seconds for a drop of 100 cm.
- The timer/dropper should
**watch the floor**, not the ball, when dropping the ball and timing. - Note that for the highest drops the tallest team member might be necessary. Be careful. Dropping a team member in lieu of a ball does not count!
- The stopwatch is measuring 100ths of a second.
**00 00 54**is actually 0.54 seconds. Note the decimal point!

Data will be recorded into a table and then plotted on an xy scattergraph, using the mean time in seconds on the horizontal x axis and the drop height in centimeters on the vertical y axis.

For data analysis a second table will be prepared using the square of the time divided by two in seconds versus the drop distance. This data will also be plotted on an xy scattergraph.

Fall time (s) | Drop height (cm) |
---|---|

0 | 000 |

100 | |

200 | |

300 | |

400 | |

500 |

Use your calculator to calculate the square of the times divided by two (t²/2) in the table above and record the results below.

(fall time²)/2 [x] (s²) | Drop height (cm) [y] |
---|---|

0 | 000 |

100 | |

200 | |

300 | |

400 | |

500 |

This laboratory report will have two tables and two graphs.

Make an xy scattergraph for first table. Graph the time t versus drop height data. The data should be slightly non-linear. Add a power regression to the data to fit a smooth curve to the data. If the theory is correct, then the graph should be a gentle curve with a parabolic shape. Remember to include the units in the header cells of the table. Do not put units in the data cells of the table in a spreadsheet. The "letters" will cause a spreadsheet to fail to graph the data as xy scattergraph data.

Graph the (time²)/2 versus the drop height. If the theory holds true, then this data should plot **roughly** as a line. Insert a linear trend line to find the slope. Use a spreadsheet to generate this graph and then copy and paste the graph into a word processor for your report.

The slope m is the experimental acceleration of gravity g.

On the graph the rise is centimeters and the run is seconds². Slope is rise over run. Therefore the units of slope and of the acceleration of gravity are cm/s².

Use WolframAlpha to determine the actual acceleration of gravity here on Pohnpei. To do this enter "acceleration of gravity 7° North 158° East" into WolframAlpha. WolframAlpha may give you the acceleration of gravity in meters per second². Multiply this value by a hundred to get cm/s².

*Demonstrate use of WolframAlpha if possible. Explain uses and applications.*

To help you with this first laboratory using WolframAlpha, the value for the acceleration of gravity g at earth's surface here at Pohnpei is 979 cm/s². To determine how close you came to this result calculate the percentage error to determine the percentage difference between your experimental acceleration of gravity g and 979 cm/s².

$\text{percentage error}=\frac{(\text{your experimental acceleration of gravity g}-979)}{979}$

Report this in the analysis [a] section of your report.

Discuss the nature of the mathematical relationship between time and distance for a falling object. Discuss whether the graph is reasonably close to a line. Report the experimental acceleration of gravity g based on the slope from the graph. Compare your result to the theoretically correct value of 979 cm/s² Discuss any problems you encountered in this laboratory including those that may have contributed to uncertainty in your measurements.

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Data | 1 Only one data point recorded correctly | 2 Only two data recorded correctly | 3 Only three data recorded correctly | 4 Only four data recorded correctly | 5 At least five data recorded correctly |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Table format | 1 Missing four table elements or format errors | 2 Missing three table elements or three format errors | 3 Missing two table elements or two format errors | 4 Missing a single table element or a single format error | 5 units in header row borders alignment of headers and data margins, table on page, repeat of headers on new page to control orphans |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Graph format | 1 Missing four elements | 2 Missing three elements | 3 Missing two elements | 4 Missing a single element | 5 Correct graph type for that lab, axis label, axis units trend line (if applicable), equation of trend line (if applicable) |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Analysis | 1 Missing four elements | 2 Missing three elements | 3 Missing a two elements | 4 Missing a single element | 5 Complete and correctly done analysis including the variable name, the units for all variables. Where appropriate, the slope, intercept, units, and meaning of the slope |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Discussion | 1 Bears little relation to the task set, unclear, very confusing, not well reasoned, extremely tangential, or extraordinarily weak. Almost incomprehensible, or a single (one to two) sentence conclusion | 2 Conclusion of little relevance to the laboratory, major gaps, or overly short such as to be incomplete, Confusing, or highly incomplete, or illogical, or made confusing by serious grammar problems, or merely restated the procedure, or a variant of a non-specific and vague "I learned a lot in this laboratory" | 3 For the most part answers the task set, though there may be gaps or redundant information, or the conclusion is essentially tangential to the experiments, or based on misconceptions, or incorrect conclusion, muddled | 4 Moderately well reasoned. Relevant and adequate answer to the task set with only a single gap or missing task item. | 5 Thoughtfully put together, well-reasoned, logical, sensible. Fully complete and thorough summary of the findings of the laboratory. Correct usage of vocabulary, appropriate use of scientific concepts. Discusses potential sources of error and how these were controlled. Includes background research on the laboratory subject. Cites appropriate text book information related to laboratory. |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Doc format | 1 Four elements of format out of compliance | 2 Three elements of format out of compliance | 3 Two elements of format out of compliance | 4 One element of format out of compliance | 5 margins, double spaced prose, no widows nor orphans, twelve point legible font, done in a word processing progam |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Grammar | 1 Very frequent errors of grammar or word order; reader often has to rely on own interpretation. | 2 Frequent errors of grammar or word order; efforts of interpretation sometimes required on reader's part. | 3 Fairly frequent errors of grammar or word order; occasional re-reading necessary for full comprehension. | 4 Some errors of grammar or word order but communication not impaired. | 5 No errors of grammar or word order. Correct use of tense. |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Vocabulary | 1 Vocabulary so limited and so frequently misused that reader must often rely on own interpretation. | 2 Limited vocabulary and frequent errors clearly hinder expression of ideas. | 3 Uses wrong or inappropriate words fairly frequently; expression of ideas may be limited because of inadequate vocabulary, or many misspelled words. | 4 Occasionally uses inappropriate terms or relies on circumlocution; expression of ideas not impaired; or a few misspelled words. | 5 Appropriate terms used consistently, clear command of vocabulary with a focus on correct usage of physical science vocabulary, no misspelled words. |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Organization | 1 Individual ideas may be clear, but very difficult to deduce connection between them. | 2 Multiple sections omitted. Little or no attempt at connectivity, though reader can deduce some organization. | 3 Multiple sections out of sequence, some lack of organization; re-reading required for clarification of ideas. For example, tables and graphs printed from a spreadsheet and then stapled to the back of a lab write-up printed from a word processing program. | 4 One section out of sequence or omitted. Material well organized; structure could occasionally be clearer but communication not impaired. | 5 All sections present in the proper order. Material exceptionally well organized. Conclusion well structured with introductory and concluding phrases. |

Criteria | Not competent | Some competency | Moderate competency | Near proficient | Proficient |
---|---|---|---|---|---|

Cohesion | 1 Communication often impaired by completely inappropriate or misused cohesive structures or vocabulary items making it difficult to make scientific sense of the conclusion. | 2 Cohesive structures or vocabulary items sometimes not only inappropriate but also misused; little sense of ease of communication. Connector words and phrases confuse and mislead the reader, but sense can be made of the conclusion. | 3 Patchy, with some cohesive structures or vocabulary items noticeably inappropriate to general style. Ideas tend to be disconnected from each other. Reads more like an outline than a coherent essay, or written as a list of answers to questions without connector words and phrases generating a choppy, disjoint style | 4 Occasional lack of consistency in choice of cohesive structures and vocabulary but overall ease of communication not impaired. | 5 Consistent choices in cohesive structures. Ideas flow logically. Conclusion remains on topic. Connector words assist the reader. |