11 March 2015. Modelling the fall of an object falling with air resistance.

Purpose

Part 1
to determine the relationship between air resistance force and speed.
Part 2
to model the fall on an object including the air resistance

Apparatus:

1. Brown coffee filter. This is going to be used as the object of the free fall that are going to be modeled  with the force of air resistance.
2. Meter stick. This will be used as the comparison to the real life measurement in the video that are going to be recorded.
3. Digital balance. to record the mass of the coffee filter.

Meter stick

Procedure:

Part 1

1. We will go to the Design Technology Building to drop the brown coffee filter from an "adequate" height to find the relationship between the filter's speed falling down, and the air resistance force.

2. As you can see in the picture, we are going to used the meter stick as a comparison measurement in the video that are recorded.
3. From this height, we are going to drop one coffee filter, then two coffee filter combined, then three, until five combination of coffee filter.

4. We are going to record these free falls in the laptop; these data later are going to be analyzed.
5. Weigh the masses of the coffee filter.

Part 2: The procedure for part 2 are combined in the Data Analysis section, since it needs data from part 1 to model the free fall object.

Data Analysis:

Part 1

• From the calculated "guessing" shown in the picture below, we now expect that air resistance force on a particular object can be modeled as F(resistance) = k.V^n.

• The V that are going to be used here is the terminal velocity, because thinking logically, the coffee filter can not add more speed (it reaches terminal velocity) because the force of air resistance is finally equal its maximum speed.
• We found each velocity of the coffee filters at a fixed height per time (s) by using logger pro as shown below.

• We acquire the terminal velocity by fitting the linear portion (indicating that the V is constant) of the position vs. time graph of each combination of coffee filters (5 combination) as shown below.

One Coffee Filter. Vt = 0.8155 m/s

Two Coffee Filter. Vt = 1.251 m/s

Three Coffee Filter. Vt = 1.385 m/s

Four Coffee Filter. Vt = 1.673 m/s

Five Coffee Filter. Vt = 1.918 m/s

• From these graphs, we can now see the slope, which is actually the value of the terminal velocity of each combination of coffee filters.
• Now, we need to find our forces acting in the coffee filters when it reaches terminal velocity, which is only influenced by gravity, which equals the weight of the coffee filters. (explanation and calculation shown below). Noted that we weighed 50 brown coffee filters = 46.3 g.

• Then, we modeled all of our forces data and velocity data into a curve fit as shown below.

• We get our equation! (pic)

Part 2: to find the model of air resistance (to find the terminal velocity), and compare it to the experimental value.

• From the data that we get, we are going to compute all of them to excel, with header of each row shown below.

• Remember, the smaller the ∆t is we are more precise in calculating the a, V, x, at any time t when we compute this to excel (explanation found in other lab post).
• Formula for each header are shown below (and how to get them).

• We use ∆t = 0.001s because we find out that when we smaller the ∆t, the results does not change much anymore, so this is already "quite good".
• We know that the coffee filter reaches terminal velocity, when a is zero, which means it does not add more speed or speed is constant.
• Below is several screen shot from excel, with varying mass of the coffee filter.

One Coffee Filter. Vt = 0.8107365 m/s

Two Coffee Filter. Vt = 1.1708268 m/s

Three Coffee Filter. Vt = 1.4516444 m/s

Four Coffee Filter. Vt = 1.6908520 m/s

Five Coffee Filter. Vt = 1.9032222 m/s

• The percent error of our experimental value of terminal velocity is shown below.

• With average percent error =

  0.2921%

  the model actually work pretty well, and the error can actually be caused by the wind from other direction blowing up the coffee filter, the uncertainties of the mass of the coffee filter, small mistakes in plotting the direction of free fall, and other external factors that we did not calculate.

Conclusions:

• Using this model that we found, we can calculate any objects with variety of mass and find it terminal velocity.
• Our model work pretty well with percent error of 0.2921%.
• Force of air resistance in the Building Design and Technology can be modeled as: