Air Resistance Experiment

Objectives
To compare the motion of and forces on a falling cupcake cup with those for a falling ball. To construct theoretical models of falling cups based on Newton's second law. To determine whether air resistance is a viscous or a drag force by comparing the models with videos of real falling cups.

What to Use
Cupcake cup, ball.

What to Do
Work with one or more partners on this experiment. As you experiment, draw your sketches, force diagrams and graphs in your notes or on a whiteboard, and discuss them with your partners and/or instructor. When you have finished this activity, you should be able to answer the following questions:

1. How and why does the motion of a dropped cupcake cup differ from that of a dropped ball?
2. What is the difference between a drag force and a viscous force?
3. Is air resistance a drag force or a viscous force?
4. How does the terminal velocity depend on mass?

1. Background
Air resistance, like sliding friction, is a "retarding" force that acts opposite to an object's velocity. But unlike friction, the air resistance force depends on the object's speed. There are two types of forces that have this property:

1. the viscous force, which depends linearly on v
2. the drag force, which depends on v2.

Both the viscous and drag forces also depend on the object's frontal area (cross-section perpendicular to its velocity).

2. Compare the motions of a falling cup and ball
Drop a cupcake cup from rest and watch it fall to the ground. Do the same with a ball that is much heavier but has approximately the same frontal area as the cup (a tennis ball is a good choice). Repeat as often as you wish. Observe carefully. Describe and discuss the motions with your partners--for each, how does the velociy change qualitatively with time?

3. Predict velocity-time graphs
Sketch a predicted velocity-time graph for the ball falling from rest and compare with your partners' sketches. Then sketch a predicted velocity-time graph for the cup. What is the slope of the v-t graph when the cup has reached terminal velocity?

4. Sketch force diagrams
Sketch force diagrams for (a) the ball and (b) the cup at the instant of release (i.e., at rest). Compare with your partners' sketches. Do they both experience the same weight force? Do they both experience the same initial acceleration?

Sketch a force diagram for the cup when it has reached its terminal velocity. Compare with your partners' sketches. Then sketch a force diagram for the ball when moving at the cup's terminal velocity. (Note: the ball and cup are moving at the same speed in these two diagrams.) Do they both experience the same air resistance force at this speed?

5. Track a cup and compare the velocity-time graph with your sketch
Complete the Terminal Velocity Tracker exercise. Be sure to save your work in a Tracker data file.

How well does the vy-t graph for the mass 5 cup compare qualitatively with your predicted graph (i.e., does it have the same shape)? Based on the graph, what is the cup's speed at terminal velocity?

6. Model the behavior of a falling cup experiencing a viscous or drag force
Complete the Dynamic Modeling Tracker exercise. Be sure to save your work in a Tracker data file.

Can you get the drag model to match the real cup's motion as well as the viscous model does? Can you tell which of the two force types is a better model for air resistance?

7. Compare the viscous force model with the motions of lighter and heavier cups
Complete the Viscous Model Tracker exercise. Be sure to save your work in a Tracker data file.

Can you get the viscous force model to match the motions of all three cups reasonably well?

8. Compare the drag force model with the motions of lighter and heavier cups
Complete the Drag Model Tracker exercise. Be sure to save your work in a Tracker data file.

Can you get the drag force model to match the motions of all three cups reasonably well? Can you now tell which of the two force types is a better model for air resistance?

9. Derive an equation for terminal velocity as a function of mass
Check that you and your partners agree on which of the two force types is a better model for air resistance. Write down the equation for the force y function for that model.

When a falling cup reaches terminal velocity, the net force acting on it is zero (why?). Using this fact, solve your force y equation for the terminal velocity as a function of mass. What terminal velocity does your equation predict for the mass 1 cup? How about for mass 4? How is the ratio of the predicted terminal velocities related to the ratio of the masses?

Open the Velocity Graphs video and study the velocity-time graphs for the falling cups. Which of the graphs correspond to the mass 1 and mass 4 cups? According to this sonic ranger data, what are the terminal velocities of those two cups? How well do they compare with your predicted values?