Pejsa's Projectile Drop Versus Distance Formula (Part 1 of 3)

Posted on 6-May-2015 by mathscinotes

Talent is God given. Be humble. Fame is man-given. Be grateful. Conceit is self-given. Be careful.

— John Wooden

Introduction

Figure 1: A Motorcycle and Rider Can Be a Projectile.

Figure 1: A Motorcycle and Rider Can Be a Projectile (Source).

I recently have had a number of readers ask me to continue my review of Pejsa's "Modern Practical Ballistics". The last major topic I have left to cover is his formula for the drop of a horizontally‑fired projectile as a function of distance.  My plan is to derive the formula and present an example of its use. The derivation is not difficult, but it is a bit long and I will divide my presentation into a couple of posts.

This post will examine Pejsa's derivation of the vertical drop differential equation.  In my second post, I will examine how he generates both an exact solution and a useful approximate solution that is commonly used in practice. My third post will contain a worked example.

I do not view Pejsa's work as the "state of the art", but it can play a useful role for people who want to develop simple applications for their ballistic work. Pejsa was working at a time when computers had limited capability and people were desperate for simple algebraic methods for getting answers. I consider his work the ultimate expression of the original empirical work done by Mayevski  and Ingalls back in the 1870s. This work was important to the development of modern gunnery and I have spent a fair amount of time reading through their papers as part of my interest in battleship gunnery.

For those who want a glimpse into modern ballistics methods, please see the work by McCoy.

Background

Approach

I am going to work hard to use Pejsa's notation, which I do not like and he applies inconsistently. However, using his notation will make referring back to his book much less confusing. I will do my best to carefully comment on the role of each equation and how I am using it.

Definitions

n
Atmospheric drag at velocities below the speed of sound is often modeled as varying by the square of the projectile velocity. n is a correction term that is used to "fix" this square-law model for transonic and supersonic velocities. Pejsa defines a different n for four common projectile velocity intervals (ft/s):

  • n = 0.5 : 1400 ≤ V < 4000
  • n = 0.0 : 1200 ≤ V < 1400
  • n = -3.0 : 900 ≤ V < 1200
  • n = 0.0 : 0 ≤ V < 900

For this 3-part series, I will be focus my examples on the 1400 ft/s to 4000 ft/s interval, but the work applies equally well to the other intervals.

A
A is parameter is a proportionality constant that is specific to the projectile in question and relates acceleration to projectile velocity, i.e. \text{acceleration} = A \cdot v^{2-n}. It will be used in deriving the projectile drop differential equation, but will not play a significant role in its solution.In his book, Pejsa sometimes uses A to represent acceleration. This created some confusion for me as I read the book. In these posts, I will make sure that A is only used to represent the proportionality constant.
F(x)
Pejsa refers to F(x) this as the retardation coefficient and has units of distance. He describes his interpretation of the coefficient in the following quote.

One percent of F is distance in which a projectile loses 1% of its speed to air drag.

For a more detailed discussion of physical meaning of F(x), see this post.  F(x) and A are related by the formula F(x) \cdot A = v^n, which I will use in the mathematical development to follow.

Conventions/Assumptions

Here are the key conventions and assumptions in Pejsa's analysis.

  • lateral distance is measured along the x-axis, which is positive to the right.
  • projectile drop is measured along the y-axis, which is positive toward the ground.

    This means that projectile drop is positive in the direction of the ground.

  • the force of drag (FDrag) as a function of projectile velocity (v) can modeled using powers of velocity (2-n, where n varies with the velocity of the projectile), i.e. {{F}_{{Drag}}}\propto A\cdot {{v}^{{2-n}}}.

    Newton showed that the drag on a slow-moving projectile (i.e. less than transonic speed) varies with the square of it velocity. The drag on projectiles moving at transonic and supersonic speeds can be modeled using other powers of velocity. The convention as been to define n as the correction required to the low-velocity, square-law model.

  • I use a yellow highlight for key results from Pejsa's book and a green highlight for significant intermediate results.

Analysis

Tools

All the work done here was in Mathcad, which for this post was primarily used as a mathematical editor. I will use its numerical capabilities in my second post on this topic.

Pejsa's Vertical Drop Differential Equation

We are going to derive the following differential equation for the projectile's velocity in the y-direction as a function of distance.

\displaystyle \frac{{d{y}'}}{{dx}}+\frac{{A\cdot {y}'}}{{{{v}_{0}}^{n}\cdot \left( {1-\frac{{n\cdot x}}{{{{F}_{0}}}}} \right)}}=g\cdot {{\left( {1-\frac{{n\cdot x}}{{{{F}_{0}}}}} \right)}^{{-\frac{1}{n}}}}

where

  • g is the acceleration due to gravity.
  • F0 is F(0).
  • n is an exponent determined by the projectiles' velocity. n is a constant in certain  velocity intervals.
  • v0 is the initial projectile velocity.
  • A is the retardation coefficient, which is a function of the ballistic coefficient.
  • y' is projectile velocity in the y-direction.

Application of Newton's Laws of Motion

Figure 2 shows the Free Body Diagram (FBD) of the projectile and the associated differential equations.

Figure 2: Differential Equation Setup.Figure 2: Differential Equation Setup.

Figure 2: Differential Equation Setup.

Equation 5 in Figure 2 is the differential equation for the projectile drop as a function of time. We want to modify this differential equation so that we can solve for the projectile drop as a function of horizontal distance.

Velocity Versus Horizontal Distance

In Figure 3, I show how Pejsa computes the projectile velocity as a function of distance.

Figure 2: Derivation of Velocity Versus Horizontal Distance.

Figure 3: Derivation of Velocity Versus Horizontal Distance.

Derivation of Flat Trajectory, Projectile Drop Differential Equation

Figure 4 shows how to derive Pejsa's differential equation for the drop of a projectile fired on a flat trajectory as a function of horizontal distance x.

Figure 4: Key Differential Equation with Closed Form Solution.

Figure 4: Key Differential Equation with Closed-Form Solution.

Equation 11 in Figure 4 is the key result. This form of the differential equation is useful because:

  • It gives us projectile drop as a function of distance.
  • It will be shown to have a closed-form solution.
  • All parameters can be determined using a projectile's ballistic coefficient, which is  readily available for different projectiles.
  • It can be used over a wide range of projectile velocities by changing the value of n in the equation.

Conclusion

Given the key differential equation, we can generate a closed-form solution in our next post. If you want to know a bit about Arthur Pejsa, here is a Youtube interview with him.

Posted in Ballistics | 13 Comments

I Must Be Old, All the Bugs Sound Familiar ...

Posted on 4-May-2015 by mathscinotes

An expert is a man who has made all the mistakes which can be made in a very narrow field.
— Niels Bohr, Nobel Prize winning physicist

Figure 1: Boeing 787 Dreamliner.

Figure 1: Boeing 787 (Wikipedia).

I was reading an article this morning about a bug discovered in Boeing's 787 software that will occur after 248 days of continuous operation. The moment I read "248 days", I let out a sigh – I have seen that number before.

Nearly every system I have worked on encountered a bug that occurred if the system was left on longer than a given period of time. In my case, the number was 248.551 days. It was caused when a 32-bit integer that was counting system running time in 100ths of a second overflowed. Figure 2 shows how you can calculate that number.

Figure 2: Calculation of 248 Day Deadline to Overflow.

Figure 2: Calculation of 248 Day Deadline to Overflow.

I read about this type of bug regularly. For example, during the Gulf War, the Patriot missile system had a bug that rendered it ineffective after 100 hours of continuous operation. While the details of this failure are different (24 bit integer, counting 10ths of seconds, failure actually caused by inaccurate type conversion), the outcome was system failure. An Arianne rocket also experienced a vehicle loss because of an integer overflow.

Boeing is encountering problems that many other engineers have experienced. For example, like Boeing, I also have dealt with lithium battery issues (see this post). While engineers generally do not make the same mistakes over and over, there are enough differences between systems that previous lessons must be modified to apply them correctly to new situations. This is where the problems occur.

I remember reading years ago that NASA would have independent teams developing software that they would run on different computers and then they would have the computers vote on the correct answer. They soon found out that people writing software tend to make similar errors. The same is true for hardware engineers.

Posted in software | 2 Comments

Good Example of the Economy of Scale

Posted on 28-April-2015 by mathscinotes

Censorship reflects a society's lack of confidence in itself.

— Potter Stewart

Introduction

Figure 1: JLC Graph of Deck Building Costs Per Square Foot As a Function of Total Square Footage.

Figure 1: JLC Graph of Deck Building Costs Per
Square Foot As a Function of Total Square
Footage (Source).

I have spent much time during my career collecting data on how production costs vary with production rate. For example, I have good data to support that unit optical component costs have dropped by 7% for every doubling of quarterly use rate (discussion). The functional relationships between product cost and volume varies depending on the type of product.

The simplest functional relationship I see is with lead-acid batteries. For most quarterly purchase quantities, the cost of a standard lead-acid battery is strongly dependent on the commodity price of lead.  Plastic enclosures show a similar cost dependency on the cost of petroleum.

Yesterday, I saw an interesting article in the Journal of Light Construction (JLC) that does an excellent job of showing how the per-square foot cost of  a deck construction drops as the total area of the deck increases (Figure 1). I found this interesting because I am going to have a structure built on my lake property in northern Minnesota. I thought this data would provide a good illustration of how I use Mathcad generate product cost models.

Analysis

Raw Data

Figure 2 shows the raw data for deck cost per square foot as a function of total deck area.

Figure 2: Plot of Raw Data.

Figure 2: Plot of Raw Data.

Figure 2 shows that the cost per square foot reduces dramatically as the area of the deck increases. The functional relationship looks nearly linear on a log-log plot, but there could be a bit of curve flattening for large deck areas.

Curve Fitting

Figure 3 shows my curve fitting exercise for two different functions: (1) a standard power law relationship, and (2) a power law relationship with offset (aka constant). The power law with offset looks like a better fit and is what I would use.

Figure 3: My Trial of Two Different Curves.

Figure 3: My Trial of Two Different Curves.

Curve Fit Examples

Total Cost

Figure 4 shows how the total cost of desk increases with area. You can see that the increase is linear with large deck sizes but starts to flatten out for small decks. This is analogous to other products where buying small quantities is expensive because vendors must charge for a fixed amount of overhead per transaction and there are few units over which to amortize that cost.

Figure 4: Plot of Total Deck Cost Versus Square Footage.

Figure 4: Plot of Total Deck Cost Versus Square Footage.

Conclusion

I spend a fair amount of time trying to develop models for product cost. I found it interesting that construction industry has the same interests and goes through the same sorts of analysis that I do.

Posted in Construction, Management | Comments Off on Good Example of the Economy of Scale

Statistics Example from World War 2

Posted on 22-April-2015 by mathscinotes

There is only one way to avoid criticism: do nothing, say nothing, and be nothing.

— Aristotle

Introduction

Figure 1: Panther Tank, Also Known As Panzer V.

Figure 1: Panther Tank, Also Known As Panzer V. (Source)

I recently finished a course on Bayesian analysis from Statistics.com and I have been looking for application examples that will provide me with some experience using these methods. I like to compare the Bayesian solutions with the standard solutions (usually called Frequentist).

Today, I came across a problem known as the "German Tank Problem". In a nutshell, the German army during WW2 had an excellent tank called Panther and the Allies needed to know how many they would be facing after D-Day, but the Germans were not going to willingly tell them how many they had. The only information the Allies had available were some serial numbers from tanks that had been destroyed. This problem provides an excellent example of how one can derive useful information about a population given data from a relatively small sample set.

Background

Statistical Camps

There are a number of possible approaches to analyzing a statistical problem. The two approaches used in this post are:

Frequentist
A method of statistical inference that draws conclusions from sample data with the emphasis on the frequency or proportion of the data (Wikipedia).
Bayesian
A method of statistical inference that uses Bayes' rule to update the probability for a hypothesis as evidence is acquired (Wikipedia).

Problem Statement

The Allies wanted to know the number of Panther tanks they would face because the Panther was a difficult opponent to stop. Some studies have shown that multiple Sherman tanks , the Allies' primary tank, were needed to defeat a single Panther. The only information available were the serial numbers of the gearboxes of destroyed Panthers, which were known to be numbered in their order of manufacture. To estimate the number of Panthers that were produced, the Allies used the statistics of these gearbox serial numbers (uniform distribution) to estimate the maximum serial number, which equaled the total number of tanks (one gearbox per tank).

I should mention that I do not know the format of the gearbox serial numbers. In many cases, the serial numbers include the year and month of production, followed by a number that reflects sequence of production for that month. In these cases, the serial numbers could be used to estimate the monthly production rate.

For this exercise, I am just going to assume that the serial number increments by 1 for each unit built over the entire lifetime of the product.

Monte Carlo Analysis

I am going to put together an Excel spreadsheet that I will use to:

  • Generate a sequence of 10,000 numbers that represent tank serial numbers.
  • Randomly sample 100 serial numbers from the population of 10,000.
  • Apply the Frequentist and Bayesian population maximum estimators to the samples.
  • Repeat this process 100 times and plot a histogram of the results.

My plan here is to be able to see how accuracy of the maximum estimators vary over 100 sets of 100 random samples. The correct answer is always 10,000.

Analysis

Sample Generation

I performed my Monte Carlo simulation using an Excel workbook and a VBA routine that generates 100 unique random integers from the range 1 to 10,000.

I show the VBA routine that generates my serial number samples below. The routine is a modified version of a routine I saw on Ozgrid. It works as follows.

  • Create an array from  1 to 10,000 with each array element loaded with its index.
  • For each array index, swap its contents with a randomly chosen array element.
  • Pick the bottom k elements of the array as the sample set.
  • Find the maximum of the k-element sample set.
  • Return the maximum sample value to the spreadsheet that needs it.

The VBA routine is shown below.

Function RandSamples(Bottom As Integer, Top As Integer, _
Amount As Integer) As Double

Dim iArr As Variant
Dim i As Integer
Dim r As Integer
Dim temp As Integer
Dim maxi As Integer: maxi = 0
Application.Volatile
ReDim iArr(Bottom To Top)
'Generate array of all the serial numbers
For i = Bottom To Top
    iArr(i) = i
Next i
'Mix the serial numbers randomly in the array
For i = Top To Bottom + 1 Step -1
    r = Int(Rnd() * (i - Bottom + 1)) + Bottom
    temp = iArr(r)
    iArr(r) = iArr(i)
    iArr(i) = temp
Next i
'Take your samples from the bottom k array indices.
'Find the maximum of your sample set.
maxi = -1
For i = Bottom To Bottom + Amount - 1
    maxi = Application.WorksheetFunction.max(maxi, iArr(i))
Next i
RandSamples = maxi 'Return your sample maximum.

End Function

Frequentist Approach

The Frequentist approach is simple. You can derive a minimum-variance, unbiased estimator that is a formula that estimates the population maximum by assuming the sample maximum is biased below the population maximum by the average spacing of the samples.

Using this approach, the Frequentist estimator of the population maximum is shown in Equation 1.

Eq. 1 \displaystyle N=M+\underbrace{{\frac{{M-m}}{k}}}_{{\text{Average Spacing Between Samples}}}=M\cdot \left( {1+\frac{1}{k}} \right)-m

where

  • N is the estimate for the largest member of the population (i.e. the number of tanks).
  • M is the maximum sample from the sample set.
  • m is the minimum sample from the sample set.
  • k is the number of samples in the sample set.

Figure 2 shows a histogram of the Frequentist population maximum estimator (Equation 1) for 100 sets of 100 samples.

FIgure M: Distribution of 100 Maximum Estimates Using the Frequentist Model.

Figure 2: Distribution of 100 Maximum Estimates Using the Frequentist Model.

Bayesian Approach

The Bayesian population maximum estimator formula is shown in Equation 2. You can see the derivation here.

Eq. 2 \displaystyle N=\left( {m-1} \right)\cdot \frac{{k-1}}{{k-2}}

Figure 2 shows a histogram of the Bayesian population maximum estimator (Equation 2) for 100 sets of 100 samples.

FIgure M: Distribution of 100 Maximum Estimates Using the Bayesian Model.

FIgure M: Distribution of 100 Maximum Estimates Using the Bayesian Model.

Conclusion

This was a good example of how seemingly unimportant information (serial numbers) can be used to generate useful information. I have seen this approach used in a number of places to estimate the production of products, like iPhones.

This technique is now well-known enough that people are even talking about encrypting serial numbers to prevent competitors from deriving sales information on their products.

Posted in Statistics | Comments Off on Statistics Example from World War 2

Banana Equivalent Dose

Posted on 17-April-2015 by mathscinotes

Let us have no more of these miserable statistics, which only paralyze the brain and freeze the blood.

— From the book Blackett's War. A plea by a British politician to not be distracted by facts.

Figure 1: Cavendish bananas, the most common desert variety.

Figure 1: Cavendish bananas, the most common
desert variety (Wikipedia, Photographer:
Fir0002).

I recently have been reading quite a bit about the hazards of traveling to Mars – one of the major hazards is radiation. This Mars reading has driven me to write a number of posts that look at the effects of radiation exposure in our daily lives here on Earth.

I wrote a post a while back on how bananas are slightly radioactive because of their potassium content. The calculations that I performed in that post confirmed the level of banana radioactivity reported in a number of places (example). However, I was not able to calculate the biological impact of this radiation because that requires knowledge of things like

  • type of radiation
  • energy radiation
  • sensitivity of the tissue to the radiation

I had no idea where to find this information, so I just computed the activity level (i.e. decay rate). I recently found an old EPA/Oak Ridge document (10 MByte) that provides some of this biological information and I thought I would use it to expand calculations to include the biological impact (called Equivalent Dose) of the banana's radiation level. This information has allowed me to confirm a number I have seen for the  Banana Equivalent Dose of 78 nanoSieverts (nSv).

Figure 2 shows the calculations. The links in the image are alive. I am not done researching this topic – I still want to know how the data in the Oak Ridge paper were derived.

Figure 2: Calculation for Banana Equivalent Dose.

Figure 2: Calculation for Banana Equivalent Dose.

Elementary Charge General Discussion Potassium in a Banana 40K in a Banana 40K in a Banana Oak Ridge Document
Posted in General Science, Geometry | 2 Comments

Candidate for the Dumbest Idea I Have Heard

Posted on 16-April-2015 by mathscinotes

If you don't think too good, don't think too much.

- Ted Williams
Figure 1: Illustration of Typical Home Plumbing Vent System.

Figure 1: Illustration of Typical Home Plumbing Vent System (Source).

My coworkers and I often discuss our home remodeling/repair adventures. This morning, a coworker described how a neighbor in his home town used a rifle to solve a common plumbing problem. I do not condone this approach and I am appalled at every possible level that someone would even try it. I think it is general knowledge that if there is an issue with the Plumbing of your home then you will 100% need to call a professional plumber in to get it sorted so that everything is up to code and there is very little room for error. That is certainly my hope anyway, so if you are reading this and believe that picking up that rifle in your home and using it for a problem like this (or any other issue) then you definitely need to take a few steps back and realize that there are a myriad of professionals out there that can help you with any issue that will crop up in your house... put down the rifle.

His neighbor had to run a vent pipe (see Figure 1, vent pipe in red) from an interior room to the roof. Running a vent pipe requires putting holes in walls and floors so that you can run the pipe to the roof. Generally, I solve this problem by taking careful measurements, drilling unobtrusive holes to verify my location, etc.

My coworker's old neighbor came up with a unique solution. Why not just shoot a rifle vertically through your house to mark the path for your vent pipe?

My mouth dropped when I heard this. Once again - call a professional, I beg of you.

Posted in Construction, Humor | 2 Comments

Another Real Life Savings Example

Posted on 15-April-2015 by mathscinotes

Beware of geeks bearing symbols and formulas.

- Warren Buffet on overly technical investing
Figure 1: Saving is Important.

Figure 1: Starting to Invest While You
Are Young is Important (Source).

My working-class upbringing did not provide me with any exposure to investing in my youth, so I really did not start actively investing until I was in my 30s. While I never look back, I do tell my sons that one thing I would do differently is starting investing right after I graduated from university. Nowadays there is a shift in how people invest, there are way more options that can be done digitally either personally or through a broker. Because of this, people are able to check on their investments using Stock signals to see what else they can invest in to help build upon the money they already have. It is a major change from what wall street used to be back in the day.

At this point, I really enjoy managing my own investments, and investing has become a family activity that I share with my adult sons. I also collect stories of amateur investors who have been successful at investing and use these stories to provide me with inspiration.

I recently saw the following answer on Quora to the question "What is the best way to make a million dollars?" This answer does a nice job of focusing on what I call "getting rich slowly" by using the power of compound interest.

I am now a multi-millionaire, and I have never had my own business, invented anything or inherited one red cent. What I have always done is paid myself first. Especially in your twenties, max out your retirement deductions, or at the minimum put in an amount that maxes out your company match. The money I contributed in my twenties now accounts for ⅔ of my net worth, even though I contributed much more each year in my thirties and later. Such is the power of compound interest!I never spent much on items that depreciate, such as cars, boats, RV's, etc. Now I am at a point where I can drive what I want. Keeping up with your family, friends and neighbors in terms of what they buy, and where they vacation is a sure way to an empty bank account. Keep in mind that many of them live beyond their means, and paycheck to paycheck. I know this is boring, however not everyone is special in their earnings potential. But anyone can end up wealthy if they follow my advice.

Posted in Financial | Comments Off on Another Real Life Savings Example

Geometric Table

Posted on 14-April-2015 by mathscinotes

Nine tenths of education is encouragement.

— Anatole France
Figure 1: Geometric Table Design.

Figure 1: Interesting Geometric Table Design (Source).

I am always looking for interesting construction projects. I don't necessarily plan to build them for myself, but I do often turn to these project for inspiration. I usually store them on Pinterest.

I occasionally mention them here because they have a mathematical slant and the table shown in Figure 1 definitely has some interesting mathematical aspects. Figure 2 shows the design's layout on a beautiful piece of wood. Something tells me that piece of wood went to a bandsaw right after the layout was drawn.

The following quote from this article on the table describes some of its mathematical aspects.

This project is a great exercise in geometry. The top and shelf are based on an equilateral triangle with circles drawn at each of the three points. An ellipse then connects each of the circles, rounding the sides as they connect to the edge of each circle.

Figure 2: Table Layout on a Piece of Walnut.

Figure 2: Table Layout on a Piece of Walnut.

Posted in Construction | Comments Off on Geometric Table

Computing a Ship's Course from Four Bearings

Posted on 13-April-2015 by mathscinotes

One machine can do the work of fifty ordinary men. No machine can do the work of one extraordinary man.

— Elbert Hubbard (1856–1915)

Introduction

Figure 1: U-boat in the Atlantic.

Figure 1: U-boats in the Atlantic (Source).

I have made no secret of my love for all things nautical – even my game playing has a nautical theme. When I have spare time, I like to play Silent Hunter 3 or 4. While these are older versions of the Silent Hunter franchise, I still enjoy playing them very much. What brings me back to Silent Hunter is how easily I can vary the level of realism to suit my gaming needs.

Recently, I have been working on increasing the level of realism by manually tracking targets. As with real WW2 submarines, this manual tracking is performed using very limited sensor data and tools:

  • Hydrophones for getting rough bearings and target speed.

    The hydrophones provided directional information and, given the ship class, you could estimate its speed by the prop rotation rate.

  • Periscope for taking optical bearings and target range

    The periscope could take very accurate bearings and it incorporated a stadimeter for estimating target range based on target height and class.

  • Maneuvering Boards

    The maneuvering board is a tool from plotting target data and can be used to derive course, speed, and position. There is a rich body of work (11 Mbyte file) on the practical applications of the maneuvering board. Here is a short Powerpoint presentation on the maneuvering board.

There is large body of work associated with determining a target's course, position, and speed using only bearings. In this post, I will focus on an algebraic technique with  geometrical options that would have been accessible during WW2. This is an interesting application of geometry and it has practical applications in other nautical situations (e.g. collision avoidance). My approach here will be to provide some background material, a high-level mathematical overview and proof based on solving a system of ten linear equations, and a worked example in Mathcad. My plan is to use the Mathcad model to generate some training scenarios for the geometrical methods that I will cover in later posts.

Ultimately, I want to be able to develop my own derivation of a totally geometric method used by gamers called Kuikueg's method, which is equivalent to the algebraic method covered here and is easily implemented used Silent Hunter's graphical interface.

Background

The world's navies have long practiced Target Motion Analysis (TMA), which is a process for determining the position, course, and speed of a target using passive sensor information (i.e. bearings with no range information). Vessels that depend on stealth, like submarines, are loath to use active sensors because these sensors give away their position – this was true even during WW2.

Having a realistic simulation of WW2 submarine combat means performing TMA to get critical target information, which we can use to maneuver the submarine into an optimum firing position and launch a "straight-runner" torpedo.

The TMA method I will be reviewing here is known as Spiess TMA (original paper, NavOps). Speiss' paper includes both algebraic and geometric methods. I will review the algebraic approach in this post and the geometric approach in a later post.

Analysis

For those who want to follow along with my Mathcad work, here is my worksheet.

Assumptions

The following analysis makes the following assumptions:

  • The target ship is moving with a constant course and speed.
  • We take three bearings at known times.
  • We setup an x-y coordinate system with the y-axis along the first bearing line.
  • We know the course and speed of our own ship.

Mathematical Overview

Mathematically, we will determine the target's course, position, and speed using the following approach.

  • Take three bearings and compute a bearing measurement at a given future time, assuming our own ship continues on its present course. We will prove that we can predict the fourth bearing measurement given the three previous bearing measurements.
  • Change the course of our ship so that we can get a different bearing from the predicted.
  • Triangulate on the target using the actual bearing and the predicted bearing.
  • Now that we have the target's course and position, we can use the three previous bearings to compute the target's speed.

The predict fourth bearing line is known as the Spiess line. Using Mathcad to solve a ten-equation linear system, I will slavishly follow the proof in Spiess' original paper to derive an algebraic equation for the Spiess line.

Spiess' paper calls out a scenario where this method will not produce a unique answer. The condition is rather complex and easily avoided, so I will not cover this situation in this post. I refer you to Spiess' (original paper) for the details.

Algebraic Derivation of Fourth Bearing Line

Figure 2 shows how the Spiess line equation is derived. Assuming that our coordinate system's y-axis is along the first bearing line, we can form a linear system of ten equations and ten unknowns:

  • 3 equations represent the bearing lines.
  • 3 equations express the target position versus along the y-axis assuming the target moves at constant speed.
  • 3 equations express the target position versus along the x-axis assuming the target moves at constant speed.
  • 1 equation puts our submarine at x = 0.

Because this is such a large system of equations, it is a bit unwieldy to show here – just click on Figure 2 to see the whole thing. I also included the solutions for the target velocity components in terms of known target information.

Figure M: Derivation of the Spiess Line Equation.

Figure 2: Derivation of the Spiess Line Equation.

Worked Example

Scenario Setup

Figure 3 shows a scenario that I setup in my Mathcad model. The model is parameterized so that I can try any scenarios that I wish. I tried dozens and they all worked, so I feel pretty confident in my knowledge of the Spiess line.

Figure M: Scenario Setup.

Figure 3: Scenario Setup.

Scenario Graphic

Figure 4 shows my graph of the example scenario.

Figure M: Scenario Chose to Demonstrate Use of Spiess Line.

Figure 4: Scenario Chosen to Demonstrate Use of the Spiess Line.

Conclusion

I was able to duplicate the work done in Spiess' original work and develop a Mathcad model that I can use for preparing some personal training scenarios. Overall, this was a good exercise. Going forward, I could use this algebraic approach to develop a Java-based tool to use while gaming, but I really want to develop my ability to use the  geometric form of this method on standard maps using compass and ruler. This will provide me with a more historically accurate gaming experience.

I will go through examples of the geometric version of this method in a later post. My plan for that post is to use Mathcad to drive Visio for generating quality graphics of some good training examples.

Posted in Military History, Navigation | 5 Comments

Negative times a Negative is a Positive

Posted on 12-April-2015 by mathscinotes

Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

— Paul Halmos on how to study mathematics
Figure 1: My Favorite Advertizing Campaign.

Figure 1: A Math Poster from My
Favorite Advertising Campaign
(Source).

On my team, I work hard to ensure that we have a non-threatening environment for questions – any questions. In fact, I often ask very basic questions in meetings so that I can make sure that I understand all the nuances of a situation. You would be amazed how often I learn things from asking questions so basic that you would think asking them would not be necessary.

Because of this approach, I frequently have non-technical people come to me and ask questions about many things, mainly remodeling and basic mathematics. I will cover the remodeling questions in other posts.

The other day I had a person come to me with the following basic mathematics question:

Why does a -1 times -1 equal a 1?

This is actually an interesting question and I last dealt with it when my kids were in school. From a mathematical standpoint, the best answer is that -1*-1 = 1 to ensure that the distributive property works, which I illustrate in Equation 1.

Eq. 1 \displaystyle 0=\text{-}1\cdot 0=\text{-}1\cdot \left( {1+\text{-}1} \right)=\text{-}1\cdot 1+\text{-}1\cdot \text{-}1=\text{-}1+1

However, this answer is not appropriate for most folks, especially schoolchildren, because they hate logical arguments and want a concrete example.

The best way to describe basic math operations is to work with analogies from their daily lives. There are numerous analogies for -1*-1=1, but for this individual the following analogy seemed to work best because they have purchased a new car.

When you buy a car, the car usually has a base price that includes a standard set of features. If you add a feature (e.g. high-end audio system), the price of the car increases. If you remove a feature (e.g. leather seats), the price of the car drops. Suppose you have selected a car with a set of features that includes price increases and decreases. What should happen to the price of a car when you remove a price reduction? It should increase. This is the same as saying that a negative times a negative is a positive.

With my kids, I used time analogies.

You hate driving, but you need to go on a long road trip to numerous locations. You assume that each destination  will add an hour of miserable driving, which you view as a negative value. If you have 10 destinations to visit, you have 10 miserable hours of driving (10*-1=-10). Suppose you remove a destination (-1*-1), now you just have 9 miserable hours of driving. Again, a negative times a negative is a positive.

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