Lab Exercise - Drag Coefficient Measurement

Goal
Edit. This experiment will allow you to find the drag coefficient of a ping pong ball. It uses a combination of physical experiment and numerical methods.
A table tennis ball can fly very fast during a game, but the air drag is a considerable factor in its flight trajectory (along with spin).
Air Drag Lab – A Fusion of Numerical Methods and Experiment
The drag coefficient was discussed in this chapter. It is a parameter that is only measurable experimentally in all but the simplest cases.  Large fluid dynamics computations can do a fair job at calculating drag for an object moving through a fluid, but ultimately the movement of turbulent fluids is still too hard to model in order to get reliable answers by calculation alone.  The way the drag coefficient is typically found is experimentally. It is measured is in a large device called a wind tunnel.  As the name suggests, the wind tunnel is a large tunnel into which an object (like a car or airplane) is placed.  Air is then forced past the object by a large fan while force sensors measure how much the object gets pushed by the air moving past it.  By measuring the force, wind speed, along with air density (easy to measure) and frontal area (also easy), one can determine the drag coefficient by using the air drag equation and solving for the only unknown parameter.     Since we don’t have a wind tunnel, we can't do that. Instead we will do the next best thing, which can be quite accurate but dangerous with automobiles and airplanes.  We will drop our object off the balcony and time its descent to the ground.  Q1: Do you expect an object experiencing drag to take more time or less time than the ideal textbook (drag-free) object to reach the ground?     After acquiring data for the fall (drop duration and distance), you will use numerical methods to model a falling ping pong ball with influences of both gravitation and air drag on the ball.  I want to mention that an analytic solution to the resulting non-linear differential equation exists in 1D, and once you’ve taken the appropriate math classes you will learn how to do this. Regardless, the equations of motion do NOT have an analytical solution in the general case of 2D or 3D motion, so for that numerical methods would be the only option.  
Procedure
  1. Measure the distance from the railing on the 3rd floor balcony to the ground to the nearest centimeter with the tape measure.  Record this.  Do it only once, correctly.
  2. Time the drop of your object from this railing to the ground 4 times with video.  Throw out the outliers if there clearly are any, and then average the time and find its standard deviation. You must know the frame rate of the video to get an accurate time. You must also get an app that will allow you to watch recorded videos in a frame-by-frame fashion so that you can literally count the frames. While this seems cumbersome, timing the drop with a stopwatch in hand introduces too much error into the data in order to get a good result in the end.  
  3. Write the drop time as t = tavg+/-1.96*(standard error).  Q2: What is your value for t written this way?
  4. Write a GeoGebra simulation that models the constant gravitational force as a vector, and the velocity-dependent drag force acting on your object. See the chapter examples if you don't know how to move forward with this step. 
  5. Adjust the drag coefficient so that the model matches reality – that is, so that the modeled object takes as long as the real one did to reach the ground.  THIS IS THE WHOLE POINT OF THE NUMERICAL SIDE OF THIS EXPERIMENT.  Without this step, there is no other way for us to find the drag coefficient.  If you think you have an alternative idea (besides the analytical solution to the 1D differential equation, see me.)
  6. By adjusting cd, have the drop time match exactly with the average drop time, and the drop distance match to within one centimeter. Q3: What are the actual numbers of the values both experimental and modeled? (Sometimes they are not identical, but they should be very close.)
  7. Now that you have this data, record cd.  You will do step 6 twice more.  Once to match t+1.96SE, and once for t-1.96SE.  Q4: What is the range of experimental drag coefficients from your model?  FYI: Sometimes measurements of quantities with errors are not symmetric about the mean value.  This is likely the case with your drag coefficient.  You might need to, for example, write it like this: cd = 0.63 (+0.09/-0.13).
Problems (Using the model)
For all of the following problems, please use the average value of the drag coefficient found in the first part of the lab unless you are being asked about a drag-free case. The idea here is the change the initial conditions of your model which were inputted in the NSolveODE command, and to ask "what-if" type questions of your model.
  1. When a drag-free object is shot vertically upward at 20m/s, how does its landing speed compare with its take-off speed?
  2. Is this true of your ping pong ball that experiences drag?  Please cite both speeds (take-off and landing) from your model.
  3. How does the rise time compare with the fall time for your ping pong ball using your mean drag coefficient?  Please cite values.
  4. Using initial conditions from #1 , what would the vy versus t plot look like without air drag? 
  5. What does it look like with air drag in your model?  Please provide the plot.
  6. A ping pong ball, if shot out of a small cannon could not easily exceed the speed of sound in air, which is 340 m/s.  If such a cannon was to shoot the ping pong ball upward at that speed, how high would it fly according to your model?
  7. What would its landing speed be?
  8. How would the rise time compare with the fall time?
– Timo Budarz

Information: Lab Exercise - Drag Coefficient Measurement