A beginner's attempt at modelling air resistance and restitution - a concerted attack on the dogma of modelling assumptions

As the title of this article may suggest, I thought it might be interesting to see what mathematical knots I could tie myself in once I begin to remove modelling assumptions from a seemingly-simple physical system. The answer, somewhat predictably, is very very many indeed.

This all stems from some work on differential equations I did over the Christmas period, where I first came into contact with the idea of modelling resistive forces as a function of a differentiated variable (acceleration usually). After doing a physics PAG (assessed practical) on determining the terminal velocity of a cupcake case falling under acceleration due to gravity in October, I realised how interesting an investigation into exactly how the object accelerates between t = 0 and terminal velocity could potentially be and with a new mathematical skillset, I began to do some scribbling...

Several months later, with some on-and-off periods of work on the paper, I have come up with this 48-page pdf file documenting how I have somewhat tangentially approached ideas such as drag and restitution with a certain level of naivety. It has not been extensively proof-read, so is most definitely an unpolished version which serves to demonstrate how I have leapt from one problem to the next, and no doubt errors will be inherent in some of my working as a result. Nevertheless, I am immensely proud of some of the derivations, as well as my real-life experiments to test them.

The following list contains all the external links to Desmos graphs, which feature throughout the document:

p13 - http://www.tinyurl.com/hmdd5jc - CTDM
p20 - http://www.tinyurl.com/jezmcxo - light-gate data
p30 - http://www.tinyurl.com/za2cq2f - multiple cycles of the CTDM in a bouncing-ball situation
p31 - http://www.tinyurl.com/hncm9su - fragility of the CTDM
p32 - http://www.tinyurl.com/grd873b - stability of the SQTM
p36 - http://www.tinyurl.com/hl32bt2 - collision impulse and force
p47 - http://www.tinyurl.com/jrmk6ye - strobe data and restitution

Introduction 

I begin the investigation with some theoretical modelling - first I take drag to be proportional to velocity, then velocity squared and finally velocity cubed. After this, I investigate how a model could be developed containing both a linear and a quadratic term in v, called the combined term drag model (CTDM). Next, I conduct some experiments with falling objects: one such experiment involves a custom-made tube of 10 phototransistor-LED light gates to observe in detail how a falling muffin-case's displacement varies with time; another determines the spring constant of the average tennis ball; another uses a strobe-light and a long-exposure camera setting to collect displacement-time data for a bouncing tennis ball. I use all this data to test the efficacy of the various models, with a good deal of running commentary and analysis.

The paper can be found here: https://www.dropbox.com/s/bgdq0vwqa0wumxp/PAPER.pdf?dl=0