Modeling the PDF of shotgun pellets at a Skeet/Trap range

Who is up for an interesting intellectual exercise?

Let's model the distribution of shotgun pellets at skeet and trap ranges.  I've started developing a model to predict the probability density solely based on the geometries involved.  

What is the probability of finding a pellet during a field sampling event? 

The image below shows the preliminary output of a single shot fired from each skeet (blue) and trap (red) position.  I've also incorporated an algorithm to randomly select the number of pellets to strike the target, thus falling out of the trajectory early in the field. 

Distribution of pellets:  A unique pellet distribution pattern was built for each shooting position.  From each skeet position, I modeled one shot trajectory for each target from the HH and LH.  This allowed me to use a Normal (Gaussian) distribution of pellets between the two trajectories, at 125 yards from each skeet-shooting position.  This was replicated for trap-shooting at 175 from the position.






How many pellets will drop early after striking the target?

Using a 30" circle at 21 yards, 85% of pellets would travel through.  This provides a general spread of the pellets as they travel from the gun bore.  Pellet density as a function of radius was calculated based on these parameters.  By comparing the pellet density at radius to the profile area of a standard clay target (4.43 in^2), I'm able to estimate the number of pellets to strike the target as a function of the distance from the center of the shot cloud.  I've created a random number generator to select the distance from the center for each shot between 0 and 30 inches, which supplies the number of pellets to strike the target. 

Then I weighted the number generator to provide more center-strikes than edge-strikes. The histogram below shows that the most strikes will drop 70 pellets.  In reality, the number of pellets that will strike a clay target are dependent upon the size, shape, and overall length of the 'Shot String'.  Here, I'm developing based on a 2D planar model of the shot string.  This complexity of a shot string is not time or cost effective at this time.  However, it seems that a range between 0 and 3% of the total pellet count would be sufficient to model the breaking zone due to the inherent complexities of the (football shaped) shot-string.

Where will these pellets land?
It was estimated that 90% of the pellet's kinetic energy would dissipate after striking the target, allowing for approximately 80 horizontal feet of additional travel for skeet-shots fired from positions 1-7, and 125 feet from position 8, and 60 feet for all trap-shots.  These distances provided a based from which distributions were developed for pellets that would fall out early from the trajectory after striking the target
The current probability distribution is coming along nicely.  I'll continue to develop this model and compare it against field evaluations.  The next steps will be to use multiple sizes of shot and incorporate potential PAH exposure from the clay targets.  Thank you all for helping to build these pieces.  The latest update includes an option for multiple/overlapping ranges.  Below is a point distribution of 4 ranges.

The Up Side of Down

"Write what you know," says Megan Mcardle, in her 2014 book - The Up Side of Down: Why Failing Well Is the Key to Success.  Positive growth in both her professional life, as well as the relationships in her personal life, represent gifts from her previous failures.  Not just a string of bad-luck, she's talking about the deep, soul-crushing periods of misery that follow the stupid mistakes, leading to sleepless nights, anxiety, dread, and regret. 

It's only after these experiences, can the wreckage of previous hopes become the foundation for something bigger and better.  We dislike failure, we're trained to by our parents, teachers, coaches, and friends.  It feels bad, which gets worse when we think others can see it.  That perception is highlighted in the example of public speaking, our collective #1 fear.  So we spend a large portion of our daily lives trying to arrange things to keep failure from happening. 

The assumption is that a failure means we've done something wrong, when the fact is that a failure is often the first result of doing something right; something we've never done before and maybe something that nobody has ever done before.  Focusing on engineering failure from our lives prevents us from embracing it, smartly.  There is a 'technocratic fallacy' in that we can remove the risk from our lives.  Failures are learning opportunities, not catastrophes. 

We can learn to identify mistakes early, so they can be corrected.  Some kinds of mistakes require punishment; others are just the natural errors that we'd expect to occur as the result of doing something we're not very good at yet.  Mistaking one for the other can be disastrous. 

My students will often hide their mistakes, cheat on an exam, or allow someone else to complete their assignments.  These are symptoms of a disastrous mechanism in education that provides no-value to our student's academic and future careers.  Failure is how we learn.  Becoming more risk-averse undermines our ability to cope with life's inevitable setbacks. 

I'll be reading this one twice, and handling out a few copies to my students.  Thank you Megan for sharing your time with us today. 

Megan McArdle is a Washington-based journalist who writes about economics, business, and public policy for Bloomberg View as a full-time blogger on July 15th. Her work has appeared in Newsweek, The Economist, The Atlantic, The New York Times, the Wall Street Journal, Time, Newsweek, and numerous other outlets.  To find more, please visit her website http://meganmcardle.com/ and follow her on Twitter (@asymmetricinfo).

Dead Men Ruling

To open the 2015 discussion series, I had the great pleasure to meet C. Eugene Steulerle about his most recent book, Dead Men Ruling: How to Restore Fiscal Freedom and Rescue Our Future. 

With the gross domestic product per household nearing $145,000 and government spending per household around $55,000 annually, we are not in a time of austerity - but one of extraordinary possibility.  We are however, constrained by a disease unique to our time - the culmination of decades of political effort to gain control over an uncertain future.  Like most chronic diseases, resources are begin wasted through misdiagnosis and treatment of symptoms: Deficits.

Deficit reduction is not enough.  It's comparable to keeping our windows and doorways open, then setting traps for unwanted animals.  Isn't it finally time to close the windows and close the doors by limiting the automatic spending increases for government programs that were designed to fulfill the legacy of politicians, now dead or long-retired. 

This leaves our newly elected representatives with an increasingly smaller amount of discretionary spending, and a lower capacity to implement the changes upon which they were elected.  Let's take a look at the Steuerle-Roeper Fiscal Democracy Index, which depicts the percentage of federal revenues remaining after mandatory spending.  Nearly all dollars are spent prior to the start of the congressional session.  This fact effectively removes decision-making capability and decreases the flexibility of our government to respond to new issues. 

Consistent misinformation, fear, and emotional reactions discourage intelligent conversation.  Let's reintroduce ourselves to the concepts of 'money', 'uncertainty', and 'long-term', to grasp a real understanding of the current fiscal climate and the opportunities that lay ahead.  Speak with your spouse, friends, colleagues, children, and students - it's their opportunity too.
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Thank you for an enlightening discussion today.  Please find more by @EugeneSteuerle from his blog, "The Government We Deserve." 

Mr Steuerle is the Richard B. Fischer Chair and Institute Fellow at the Urban Institute, and former Deputy Assistant Secretary of the Treasury, Fellow at the American Enterprise Institute and Brookings Institution, and a columnist for the Financial Times.  He serves on advisory panels for the Congressional Budget Office, Government Accountability Office, and the Joint Committee on Taxation.  An engaging man, and outstanding scholar - thanks for stopping in.