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Cliff Diving and Quadratic Drag
Developed by Peter Bryant  Published May 31, 2019
This exercise set focuses on the estimation, modeling, and experimental verification of the depth of water required to bring a cliff diver to a near stop. Students use dimensional analysis to "derive" the dependence on velocity, density, and area of hydrodynamic drag, and then they use energy principles to determine the amount of energy that must be dissipated by drag, which leads them to an estimate of the depth of water required. By dropping a nearly neutrally buoyant ball into a glass container filled with water, they can verify or eliminate their models. Students use a spreadsheet or a loop in a procedural language to solve for the motion using the Euler method.
Subject Areas  Mechanics and Experimental Labs 

Level  First Year 
Available Implementations  IPython/Jupyter Notebook and Spreadsheet 
Learning Objectives 
Students who complete the exercises will be able to:
* determine an expression for hydrodynamic drag using dimensional analysis. (Exercise 1)
* apply energy methods to hypothesize the depth of water needed to stop a dropped ball. (Exercise 2)
* use experimental techniques to verify or rule out a model. (Exercises 2, 3, and 5)
* develop and implement an algorithm for solving for motion under quadratic drag. (Exercises 4 and 5)
* produce and analyze graphs of computational solutions. (Exercises 5 and 6)
* asses the accuracy of computational solutions by comparing with analytical solutions. (Exercises 5 and 7)

Time to Complete  150 min 
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
#### Exercise 1: Drag in Water
Using dimensional analysis and your common sense, determine a formula for *hydrodynamic drag*, or the drag force in water. Recall that $1 \textrm{N}=1 \frac{\textrm{kg}\ \textrm{m}}{\textrm{s}^2}$. The complete formula contains dimensionless constants that you won't be able to guess, so I'll give you those when you're finished.
**Optional Prompt:** Steps for dimensional analysis:
1. Determine the combination of fundamental units (units for mass, length, and time) that the formula must have in order to give a force.
2. Determine the physical quantities in your system upon which the drag force *can possibly* depend.
3. Try different arrangements of the relevant physical quantities until you find equations that give correct units for the drag force. You may find more than one option.
4. To determine if your formula makes physical sense, vary the physical quantities oneatatime. Will your formula predict the correct behavior as the physical quantities become very small and very large?
**Optional Website with Review:** http://farside.ph.utexas.edu/teaching/301/lectures/node8.html
#### Exercise 2: Stopping Distance (Static Estimate)
The work done by hydrodynamic drag is equal to the change in thermal energy, $\Delta E_{th}$, of the closed diverwater system. Use this fact and the conservation of energy to estimate the minimum depth of water required to stop a diver jumping from a height $h$ above the water's surface.
You may assume that a diver, being largely composed of water, is nearly neutrally buoyant. This means that you may ignore the diver's weight and thus any contribution from the gravitational potential energy after he or she enters the water.
**Optional Instruction, which may be omitted for a more discoverybased activity:**
Depending on your approach, you may need to assume a speed in order to calculate the drag force. For now, make a static approximation and use for the speed of the ball the speed at which it enters the water. We'll refine this approximation later.
#### Exercise 3: Experimental Verification
Fill a glass cylinder with water and drop your rubber ball from a height of 50 cm above the surface of the water. Measure the distance it takes the ball to stop sinking, and include an uncertainty.
Apply your model from **Exercise 2** to the case of the dropped ball. Is your measured distance consistent with your prediction from energy conservation?
**Optional (together with the optional part of Exercise 5):**
Record video of one of your your drops and analyze it framebyframe to make a plot of the ball's depth as a function of time since it first hits the water.
#### Exercise 4: Hand Calculation of Stopping Distance (Dynamic Model)
The problem with the static estimate is that we have assumed that the ball maintains a constant speed and thus a constant drag force as it slows down. In reality, as the ball slows down, the drag force decreases (why?). We have a situation where work is done by a nonconstant force, and the interdependence between speed and force goes something like this:
1. The ball enters with a high speed.
1. The water uses a high drag force to slow the ball down.
1. As the ball slows down a bit, the drag force gets slightly smaller.
1. The water uses this slightly smaller drag force to slow the ball down a bit more.
1. As soon as the ball slows down a bit more, the drag force gets even smaller.
1. Repeat...
To account for the changing drag force, we'll break the water into thin, horizontal layers, over each of which we can assume the velocity and thus drag force are constant. If you know the ball's speed in a layer, then, you can calculate the drag force for that layer. Given the thickness of the layer, you can calculate the work done by drag over that layer, which tells you how much Thermal Energy increases and thus how much Kinetic Energy the ball will lose as it travels through the layer. From that, you can calculate the speed of the ball in the layer below. Then repeat for the next layer. The image below is a schematic of what you can calculate in each layer.
![](images/Diving/slice_flowchart.png "")
Your task is to use this approach to calculate by hand the speeds in the first four layers of thickness $d=0.001$ m, starting at the top of the water. Assume the speed in the first layer is just the speed at which the ball hits the water.
#### Exercise 5: Computation of Stopping Distance
Use a spreadsheet (or a programming language) to automate the procedure you just did by hand. Extend the calculation to a final depth a bit beyond what you observed in your experiment, and make a plot of speed as a function of depth. How does the dynamic model compare to the measured maximum depth from your experiment?
What happens if you change the thickness of every slice to $d=0.01$ m?
To $d=0.0001$ m?
Should $d=0.01$ m, $0.001$ m, or $0.0001$ m give you the best approximation? Why?
**Optional (together with the optional part of Exercise 3):**
Assuming the ball's speed is constant as it travels through a slice, on your spreadsheet calculate for each slice the time it takes for the ball to pass through it. From that, make a plot of the ball's depth as a function of time, and compare it to the measured result in **Exercise 3**. Is there a point at which you would say the model no longer accurately reflects the position of the ball? Explain.
#### Exercise 6 (optional): Deeper Analysis and Buoyancy
You'll note that in our computational solution, the ball continues to slow down but never actually comes to a stop. What are some things we omitted from our model that would cause the simulated ball to stop sinking had we included them?
#### Exercise 7 (optionalrequires calculus): Theoretical Treatment
1. Argue that, based on our assumption that the ball is neutrally buoyant, in the water $$\Delta KE =  \Delta E_{th} =  F_D\Delta y ,$$
where $F_D$ is the force of hydrodynamic drag and $y$ is positive downward.
2. Let $\Delta \rightarrow d$ to go to the infinitesimal limit, and use your equation for the hydrodynamic drag from **Exercise 1** to find and solve an integral equation relating the depth in the water, $y$, to the speed, $v$. Plot your equation over your results from **Exercise 5** and compare.
*Hint:* use $d(KE)=d(mv^2/2) = m\ v\ dv$.
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Credits and Licensing
Peter Bryant, "Cliff Diving and Quadratic Drag," Published in the PICUP Collection, May 2019.
The instructor materials are ©2019 Peter Bryant.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license