“Come and listen to a story ’bout a man named Jed
Poor mountaineer barely kept his family fed
Then one day he was shooting for some food,
And up through the ground come a bubbling crude
(Oil that is, black gold, Texas tea)

Well the first thing you know old Jed’s a millionaire
Kin folk said Jed move away from there
Said California is the place you oughta be…” 
—The Ballad of Jed Clampett

Oil and natural gas have been important in the US and globally for nearly 180 years, with that importance spiking following invention and widespread adoption of the internal combustion engine in the 19th and 20th centuries.  Beginning in 1973, the US began consuming more crude oil than could be domestically produced.  Dependence on global markets made the US economy vulnerable to oil-price shocks in the 1970s, and acted as one of several proximate triggers for the 1990-1991 recession.

In order to better understand production and labor dynamics in US oil and gas production over time, I examined and downloaded production data from the Energy Information Administration (US DoE) and oil and gas employment data from the Bureau of Labor Statistics (US DoL).  Graphs of the raw data are below:

Natural Gas Withdrawals and Production in the United States, Jan 1980 through Sept 2016.

US Field Production of Crude Oil, Jan 1920 through Sept 2016.

Oil and Gas Employment in the US, Jan 1972 through Nov 2016.

A few points stand out when examining the data:

• Natural gas production reached a bottom in the mid-1980s, rising modestly until the mid-1990s, and remained stable until the mid-2000s.  Production is currently just off the 2014-2015 peak.
• Crude Oil production in the US peaked in the early 1970s, though oil and gas employment peaked around 1980.  Since then, a second production peak occurred in 2015, though production has fallen off slightly since then.
• In spite of recent peaks in production, employment in oil and gas extraction is about 35% off the 1982 peak of 266 thousand workers.

To examine oil and gas combined production and productivity, data was downloaded and opened in MS Excel.  Annual barrels of oil and millions of cubic feet of natural gas were converted to British Thermal Units (BTUs).  September employment levels were used for each year.  The total production series was divided by the employment series to calculate productivity in BTUs per worker.

Oil and Gas Extraction Employment in 1000s (top), Total Crude Oil and Natural Gas Production in 10^14 BTUs, and Productivity in 10^9 BTUs per worker, 1980 though 2016. Author’s Calculations.

The productivity data shows a nearly unbroken 20-year increase from 1983 to 2003, a modest decline, and some leveling until 2014, and then a return to productivity growth.  As one would expect, productivity increases tend to correspond to falling or stable crude oil prices.  Productivity by this measure tends to decline when prices are rising.  This makes sense because rising prices can be expected to follow periods of falling production (affecting the numerator), and lead to increases in employment (affecting the denominator).

The following is my opinion only, freely given, worth what you’ve paid for it, isn’t meant to be career or investment advice, and isn’t necessarily the view of my employer or anybody else:  The shale revolution has gone global, and demand growth is expected to be slow due to demographics and consumer preferences in the developed world. Because global supply is much less constrained than demand, I expect crude oil prices to stabilize in the 60 to 80 dollar per barrel range for the next several years, assuming no severe shocks to global supply or demand.  In order for profitability of US resources to be maintained productivity must continue to improve, implying modest but steady increases in production, and net employment changes occurring very slowly.

The Green Bay Packers and Detroit Lions played their second game of the 2015 season on December 3rd.  It ended with a highly unlikely Hail Mary catch by Richard Rodgers to score the winning touchdown.  In the days following the game I heard many, many different people expressing excitement and admiration for the play, with one colleague even declaring Aaron Rodgers “The best quarterback in the NFL”.

The league posted a video clip asking “How far exactly did Rodgers’ Hail Mary Travel?”.  The official marking of 61 yards is shown at the bottom of the screen.  This is the distance from the line of scrimmage at the start of the play to the front of the end zone line.  Ben Rohrbach expressed his belief that the total travel distance of the ball was close to 100 yards, including the vertical and horizontal elements.

Objects vaulted into the air follow a parabolic vertical path.  Credit: Plus Magazine

When a ball is thrown, kicked, or otherwise sent with force into the air, the vertical profile will be shaped like an upside down parabola.  This is because gravity acts with constant downward acceleration against the object.  The total distance along this segment of a parabola is known as the arc length.  It can be calculated by integrating the changing height across the horizontal distance. Wolfram’s discussion is here, and Had to Know’s is here.

Equations from Top: Horizontal Distance from Pythagorean Theorem; Vertical Distance calculated from hang time; Total Arclength from integration; Starting velocity of the ball neglecting drag force; Energy imparted to the ball by the passer

With that out of the way, the objective is to calculate the actual arc length of Aaron Rogers December 3rd pass.  The same method can be used to calculate golf drives, log throws, rock catapultations, etc….  Note: Air friction is neglected in the following discussion.

The horizontal distance of the pass is calculated via the Pythagorean Theorem using length and width coordinate estimates from the back end zone and visual top of the field.  The image below shows how this was done for Aaron Rodgers throw.  The throw starts an estimated 39 yards from the back end zone, and ends 103 yards from the same end zone.  At the start it’s an estimated 13 yards from the top sideline, and it ends about 18 yards from the same sideline.

Method of estimating the actual horizontal distance of a pass. Black lines demonstrate start and end distances from the passer’s back end zone, and white lines demonstrate distances from the “top” sideline.  Image Credit: Light Headed Beds

The second step is to estimate the total height attained by the ball, minus the height of the passer.  Here, I assume that the ball was caught at the same height from which it was thrown.  The vertical displacement in yards can be calculated by estimating the hang time, squaring it, and multiplying by 1.34058.

Once the vertical and horizontal displacements are measured, the rest can be calculated in a straightforward manner.  Using my estimates of pass start at (13,39), catch at (18,103), and hang time of 5.6 seconds I got the following results:

• Horizontal Displacement: 64.195 yards
• Vertical Displacement: 42.041 yards
• Total Travel Distance: 110.716 yards
• Velocity of Pass: 12.706 yards per second
• Energy Imparted to Football: 21.405 foot pounds
• Power Output of Aaron Rodgers during pass: 0.216 horsepower

This admittedly armchair analysis vindicates Rohrbach, and calculates Aaron Rodgers throwing technique to be as strong as just over a fifth of a horse.  The image of my Excel output is below, with the following cell formulas:

• A6 =SQRT(C2^2 + C4^2)
• A11 =SQRT((A6^2/4) + (4*B9^2)) + (A6^2/(8*B9))*ASINH(4*B9/A6)
• A13 =(3*B11)^2*(0.94799/(2*32.174049))
• B9 =1.34058*A9^2
• B11 =SQRT((A6/A9)^2 + 10.72467*(A9/2))
• B13 =(A13/B6)/550
• C2 =ABS(B2-A2)
• C4 =ABS(B4-A4)