**Objective:** To determine the number of coffee filters present in a trial once you know the magnitude of their terminal velocity.

**Background Information:** The common mathematical formula for distance is

*d = rt* where the rate,

*r*, represents the average speed. Since the filters in this

experiment will travel in only one direction - straight down - their average

speed will equal the magnitude of their average velocity. In subsequent formulas on this page,

*r* will be replaced with

*v*.

In this

experiment the falling coffee filters will be experiencing two basic forces:

- weight (the product of mass and gravity), and
- air resistance.

Any object that is falling at a constant, terminal,

velocity is experiencing dynamic equilibrium and can be modeled with the

equation . In this equation:

**Materials Needed:**- CBR
- Coffee filters (same size of any size filter)
- Graphing calculator

**Length of Activity:** Two class periods working in groups of three. One person will drop the filters, one person will hold the CBR and collect the

data points, and one person will be the

data recorder.

**Procedure: PART I - Data Collection**- Set up the graphing calculator and CBR.
- Drop one filter, choose two points on the linear section of the distance graph. Choose one point very close to the start of the linear segment and one at the end. Record these points in the chart. Repeat for a total of five trials.
- Repeat step 2 with stacks of two filters, three filters, four filters, and five filters, but you only need to run three trials with each of these combinations. Record all data in the charts.
- After you have recorded all of the data points, fill out the remainder of the charts. Calculate the average velocity for each number of filters by averaging the final "average velocity" column in each chart.

**One Filter**Trial | t_{1} time_{1} | y_{1} position_{1} | t_{2} time_{2} | y_{2} position_{2} | s = y_{2} - y_{1} net displacement | Dt = t_{2} - t_{1} time interval | v = s/Dt_{ } av velocity |

1 | | | | | | | |

2 | | | | | | | |

3 | | | | | | | |

4 | | | | | | | |

5 | | | | | | | |

Average velocity for one filter = ___________

**Two Filters**Trial | t_{1} time_{1} | y_{1} position_{1} | t_{2} time_{2} | y_{2} position_{2} | s = y_{2} - y_{1} net displacement | Dt = t_{2} - t_{1} time interval | v = s/Dt_{ } av velocity |

1 | | | | | | | |

2 | | | | | | | |

3 | | | | | | | |

Average velocity for two filters = ___________

**Three Filters**Trial | t_{1} time_{1} | y_{1} position_{1} | t_{2} time_{2} | y_{2} position_{2} | s = y_{2} - y_{1} net displacement | Dt = t_{2} - t_{1} time interval | v = s/Dt_{ } av velocity |

1 | | | | | | | |

2 | | | | | | | |

3 | | | | | | | |

Average velocity for three filters = ___________

**Four Filters**Trial | t_{1} time_{1} | y_{1} position_{1} | t_{2} time_{2} | y_{2} position_{2} | s = y_{2} - y_{1} net displacement | Dt = t_{2} - t_{1} time interval | v = s/Dt_{ } av velocity |

1 | | | | | | | |

2 | | | | | | | |

3 | | | | | | | |

Average velocity for four filters = ___________

**Five Filters**Trial | t_{1} time_{1} | y_{1} position_{1} | t_{2} time_{2} | y_{2} position_{2} | s = y_{2} - y_{1} net displacement | Dt = t_{2} - t_{1} time interval | v = s/Dt_{ } av velocity |

1 | | | | | | | |

2 | | | | | | | |

3 | | | | | | | |

Average velocity for five filters = ___________

**Procedure: PART II - Data Analysis**- As a class, determine the average mass of one coffee filter by massing 100 filters and dividing the result by 100.
- Using the mass and average velocity for one filter, calculate the value of
*b* in *mg = bv*^{2}. Remember that g = 9.8 m/sec^{2} and that the mass of the filter must be measured in kg. - Using this value of
*b* as a constant that does not change, use *mg =bv*^{2} to determine the mass of each stack of coffee filters. - Dividing this total mass by the average mass of one coffee filter you can experimentally determine the number of filters,
*N*, used in each trial. - Calculate the relative error in each of your
*N* values compared to the number of filters in each stack. Remember: *relative error = |actual value – experimental value|* - Calculate the percent error for each trial.
*percent error =*** 100*

| 1 filter | 2 filters | 3 filters | 4 filters | 5 filters |

Total Mass | | | | | |

*N* | 1 | | | | |

Relative error | 0 | | | | |

Percent error | 0% | | | | |

**Procedure: PART III - Conclusions**Answer the following questions based on your measurements.

- Were the
*N* values you found exactly equal to the number of filters in each stack?

- Is the percent error approximately constant for each group of filters? Or, does the error increase or decrease as
*N* gets larger?

- Assuming that the mass you were given is correct, what do you think could account for these errors?

**Extension:**Using the data from all five combinations, find a relationship between the average velocity, total mass, m, and the number of filters, *N*. That is, can you predict the average velocity of 8 filters?