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.

*Instructional note. This laboratory builds on two earlier activities that during a regular school usually occur on Monday and Wednesday respectively. The first activity was a plot of time versus distance for an accelerating RipStik. Based on the RipStik activity, there was a non-linear relationship (curved line) between time and distance for the accelerating RipStik. In the second activity, the arc of a ball activity, the students explore whether the trajectory of a ball might be related to a parabola. A quadratic equation is presented as the underlying mathematical relationship. The students graph their data and the equation to explore whether the ball arc and the equation are related. These two activities build to this laboratory where the specific mathematical relationship nature of time versus distance for a falling ball is measured. Put more simply, if a jacks ball falls twice as far, by what factor does the time increase? Ball arc activity information.
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- Explore the nature of the mathematical relationship between time and distance for a falling object
- Determine a value for a non-linear mathematical relationship
- Determine the acceleration of gravity g

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 linearily. 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.

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

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 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 20 centimeters and drop the ball from 120 cm.
- Move up by another 20 centimeters and drop the ball again. Continue going up by 20 centimeters until you reach 300 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 measurments. 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 graph paper, 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 in seconds versus the drop distance. This data will also be plotted on a graph sheet.

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

0 | 000 |

100 | |

120 | |

140 | |

160 | |

180 | |

200 | |

220 | |

240 | |

260 | |

280 | |

300 | |

400 | |

500 |

** After completing the data up to 300 centimeters, groups will work with the instructor to attempt to gather data at 400 cm and 500 cm using the balcony. We will have to work quietly and carefully when working outside.*

*Do not graph this data.*

Use your calculator to square the fall times in the table above and record the results below.

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

0 | 000 |

100 | |

120 | |

140 | |

160 | |

180 | |

200 | |

220 | |

240 | |

260 | |

280 | |

300 | |

400 | |

500 |

Graph only 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.

The "textbook" value for the acceleration of gravity g at earth's surface is 980 cm/s². How close did you come to this result? Calculate the percentage error to determine the percentage difference between your experimental acceleration of gravity g and the value quoted in science texts. Report this in the analysis [a] section of your report.

$\text{percentage error}=\frac{(\text{your experimental acceleration of gravity g}-980)}{980}$

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 "textbook" value of 980 cm/s² Discuss any problems you encountered in this laboratory including those that may have contributed to uncertainty in your measurements.

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Some calculators can perform a linear regression. Your instructor might choose to assist groups with determining the slope and intercept for their data using their calculators.
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