# 14.6: Unit 8 Lab Extension- Modeling Terminal Velocity and Extracting Drag Coefficient*

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## Modeling Terminal Velocity and Drag Coefficient

### Materials:

- copy of lab sheet and spreadsheet from Unit 2/3 Lab
- spreadsheet and graphing software
- for distance learners, access to online forums, videos, and help features for the spreadsheet software will likely be necessary

### Lab Objectives

The objectives of this lab are:

1) derive a physical model for terminal velocity of objects falling through air,

2) test the model against terminal velocity data

3) use our model to extract useful information from the data

### Build a Physical Model

Terminal velocity is the maximum speed reached by a falling object. Therefore, once a falling object has reached terminal velocity then it is no longer accelerating and we can say the velocity is constant, but not zero. This state is known as ___________ equilibrium.

When an object is in the state you described above, what can you say about the total force on the object?

Draw a free body diagram (FBD) of the falling object indicating the forces acting on it. Be sure to label the forces. [Hint: there are two forces].

Your diagram should show that the two forces are the same size, but pointing in opposite directions so that they cancel out. If this were not the case, then the object would not be in ____________ equilibrium. Represent this concept with an equation that sets the two forces equal:

One of the forces should be gravity (weight). Rewrite your equation, but replace the force of gravity in your equation with the formula for calculating the force of gravity on an object near the surface of the Earth:

The other force should be the drag force. Rewrite your equation, but replace that force in your equation with the formula for drag force:

Now we have a physical model for the terminal velocity. The model predicts that terminal velocity is proportional to the __________________ of the mass. Stated another way, the terminal velocity depends on mass to the _______ power.

### Hypothesis

We can turn the previous statements into a quantitative hypothesis: *If identically shaped objects of different mass are dropped under the same conditions, then the terminal velocity the objects will be proportional to the ____________ of the mass. *

### Acquire Test Data

We already have the terminal velocity data for coffee filters that we acquired during the Unit 2/3 Lab, so let’s use that. Open up the spreadsheet you created during the Unit 2/3 Lab.

### Analyze

Fit a trend line to the data. We don’t yet know what type of curve should fit, so use a power fit. This will tell us how terminal velocity depends on mass according to the data.

Write your fit equation here:_______________

Record the R^{2} value here:____________ The R^{2} value gives you an idea of how well the equation fits your data. The closer R^{2} is to one the better the fit.

Would you say that your equation fits the data well? Explain.

What power is the mass (x-variable) raised to in your equation? Is it 0.5,1, 2, 3, or something else?

### Conclusion

A power of 1 would suggest terminal velocity is proportional to mass. A power of 2 would suggest terminal velocity is proportional to mass squared. A power of 1/2 would say that the terminal velocity is proportional to the square root of the mass. Does your data support your quantitative hypothesis? Explain.

### Physical and Empirical Models

Your fit equation represents a quantitative * empirical* model. We could use the model to try to predict the terminal velocity of some other filter masses, but the model is only based on data, it doesn’t rely on any physics concepts to explain what we are observing.

Your equation for terminal velocity is a quantitative * physical* model because it allows us to predict values for terminal velocity AND it provides information about the underlying physics behind the behavior we observe.

If we were to test your two types of models many more times for many types of objects and they always did well at predicting the experimental results then we would say the models have been * validated*.

Could either of these models become part of a * theory*? Explain.

Could either of these models become part of a ** Law**? Explain.

### Extracting Model Parameters*

We have a physical model for terminal velocity, and we can quickly look up or measure all of the parameters except the drag coefficient. Combing our data and our physical model will allow us to extract that parameter value for drag coefficient.

Write down your physical model and immediately below it write down the equation you fit to the data. Compare the two equations to determine what combination of physical parameters must equal the number in front of * x* that you see in your fit equation. (Remembering that the

*in your equation represents mass and the*

**x***represents terminal velocity).*

**y**Write and equation between the parameters and the number and solve it for drag coefficient. Show your work.

Now put in the known values for the other parameters and calculate the drag coefficient, including units. Show your work below.

You have now tested your physical model against the data and used the data to extract an unknown parameter of the physical model. That is real heavy-duty science right there!