Air Rifle Math

Posted on 3-February-2014 by mathscinotes

Introduction

I received an email this weekend from a dad struggling to help his son with a project involving aerodynamic drag and BB gun. I did some quick calculations which I document here. I will try to look at pellets tomorrow. I was able to use basic principles to duplicate the empirical results quoted by manufacturers.

I will be computing three numbers associated with an air rifle shooting a BB:

  • The force of drag on the BB as it leaves the muzzle
  • The deceleration of the BB as it leaves the muzzle
  • The ballistic coefficient of the BB

Background

Definition of Ballistic Coefficient

The Wikipedia defines the ballistic coefficient as follows:

Ballistic Coefficient
The Ballistic Coefficient (BC) of a body is a measure of its ability to overcome air resistance in flight. It is inversely proportional to the negative acceleration — a high number indicates a low negative acceleration.

A projectile with a small deceleration due to atmospheric drag has a large BC. Projectiles with a large BC are less affected by drag and have performance closer to their performance in a vacuum.

While the Wikipedia definition is accurate as far as it goes, it does not allow you to compute the BC of a projectile. Equation 1 shows you how to compute the BC of any projectile.

Eq. 1 \displaystyle B{{C}_{\text{Projectile}}}\triangleq \frac{{{a}_{\text{ReferenceProjectile}}}}{{{a}_{\text{Projectile}}}}

where

  • aReferenceProjectile is the acceleration of a reference projectile (eg. G7).
  • aProjectile is the acceleration of the projectile we are interested in.

Coefficient of Drag Given Reynolds Number

Figure 1 shows the coefficient of drag graph that I digitized using Dagra for this example.

Figure 1: Coefficient of Drag Versus Reynolds Number.

Figure 1: Coefficient of Drag Versus Reynolds Number.

Density and Viscosity Data for Interpolation

Figure 2 shows the table from this web site that I interpolated so that I could get the density and viscosity of air at various temperatures.

Figure 2: Air Density and Viscosity Data.

Figure 2: Air Density and Viscosity Data.

Analysis

Interpolation of the Air Data

Figure 3 shows how I used Mathcad to interpolate all this data.

Figure 3: Digitization Code for Graphical Data.

Figure 3: Digitization Code for Graphical Data.

Calculations

Figure 4 shows how I calculated the

  • force of drag on the BB
  • acceleration experienced by the BB
  • ballistic coefficient of the BB
Figure 4: Calculations for the Drag and Ballistic Coefficient of a BB.

Figure 4: Calculations for the Drag and Ballistic Coefficient of a BB.

Table of Viscosities Table of Densities Diameter of a BB Graph of Drag Coefficients Vs Reynolds Number Wikipedia article on drag coefficients Wikipedia article on drag coefficients Reference for Ballistic Coefficient of a BB Reference for mass of a BB Reference on types of ballistic coefficient

Conclusion

I computed three numbers associated with an air rifle shooting a BB:

  • The force of drag on the BB as it leaves the muzzle: 0.17 N = 0.038 pound = 0.613 ounce

    I cannot find corroborating information on the web. However, I use this number to compute the ballistic coefficient, for which I do find corroborating evidence.

  • The deceleration of the BB as it leaves the muzzle: 1691 ft/sec2

    Straightforward application of Newton's second law.

  • The ballistic coefficient of the BB: 0.014

    This agrees with data I have seen on forums here and here.

Next, I will look at a 22 caliber pellet.

Posted in Ballistics | 5 Comments

Proofs Without Words

Posted on 1-February-2014 by mathscinotes

Quote of the Day

If you've been playing poker for half-an-hour and you don't know who the patsy is – your're the patsy.

— Warren Buffet

I love proofs without words. Usually, I see them in discussions of mathematical concepts like the Pythagorean theorem (examples). While reading about home construction at the Journal of Light Construction, I saw a nice graphical look at the difference between the gross margin on a product and the markup on a product.

Posted in General Mathematics | Comments Off on Proofs Without Words

At What Fuel Tank Level Should I Refuel?

Posted on 1-February-2014 by mathscinotes

Quote of the Day

If winning isn't everything, why do they keep score?

— Vince Lombardi

Auto_fuel_gaugeIt is cold in Minnesota right now and I hear the local motorheads warning people not to run their gas tanks down to empty. During winter, I hear this warning because of concerns that water vapor in the gas tank may freeze and cause the vehicle to stop. I also hear concerns about running on a low fuel tank during the heat of summer because the fuel is need to cool the fuel pump, which is often mounted in the tank. Some folks also say that there is corrosion at the bottom of the gas tank that get sucked into the fuel pump when running the tank near empty.

The recommendation is usually that people should not let their tank go below quarter full. I don't know how serious these problems are, but they sound plausible. This rule means that you have to refuel 33\%= \frac{1}{\frac{3}{4}}-1 more than someone who lets their gas tank go all the way to empty.

A software engineer who grew up on a farm commented that his father demanded that they fill their gas tanks when the tanks got down to 1/3 of a tank. His father was concerned that there could be an accident and someone may need to be rushed to the hospital. The nearest hospital was 1/3 of a full tank away. As I understand it, his father had heard of an accident where they could not get a child to the hospital in time to save him because a vehicle had been left on empty and they could not refuel it quickly enough. His father wanted to be sure that would not happen to his family. That seemed like a good reason to keep their gas tanks relatively full.

The need to fill your gas tank sometimes looms large in history. Once of the things that contributed to the demise of the battleship Bismarck (Fig. 1) was that it had not filled its fuel tanks when it had an opportunity. There were reasons not to fill her tanks, but in hindsight they were not good reasons.

Figure 1: Battleship Bismarck (Source: Wikipedia).

Figure 1: Battleship Bismarck (Source: Wikipedia).

I remember reading that US Navy fighters providing air cover for aircraft carriers (Fig. 2) never let their fuel tanks run below 70% full. Their concern is readiness -- fighters burn a lot of full when they maneuver and they wanted to be sure to have plenty of fuel in the event they need to engage a foe on short notice. This means that they need to refuel very often. They takeoff with just enough fuel to get them to a tanker aircraft -- they literally refuel right after takeoff. Reducing the weight of fuel means that they can takeoff with a larger weapons load. They then have to refuel 3.33= \frac{1}{1-0.7} times more often than if they run their tanks down to empty.

Figure 2: Fighters Flying Top Cover Over the Aircraft Carrier Kitty Hawk.

Figure 2: Fighters Flying Top Cover Over the Aircraft Carrier Kitty Hawk.

It seems like you should refuel based on the cost of needing to refuel at a bad time. If you cannot afford to refuel during an emergency, you need to keep your tank relatively full at all times. The price you pay is more frequent refueling.

Posted in General Mathematics, Personal | 3 Comments

Radiation Exposure from Cigarette Smoking

Posted on 25-January-2014 by mathscinotes

Quote of the Day

I write a lot of programs and I can't claim to be typical but I can claim that I get a lot of them working for a large variety of things and I would find it harder if I had to spend all my time learning how to use somebody else's routines. It's much easier for me to learn a few basic concepts and then reuse code by text-editing the code that previously worked.

- Donald Knuth

Introduction

Figure 1: Tobacco is slightly radioactive.

Figure 1: Tobacco is slightly radioactive. (Source)

I have been reading about the safety hazards associated with traveling to Mars. One of the hazards is radiation. Since I know very little about the biological hazards associated with radiation, I have some learning to do. One of the ways I learn about a subject is to work through problems from the various online and library references that are available. During my investigation, I came across four excellent articles on the subject of radiation exposure from smoking tobacco. So, if being unlikely to get a decent life insurance policy wasn't enough to keep you from giving up tobacco then hopefully this revelation will do the trick! This simple example illustrates the basic calculation process.

I will summarize the information here using a Fermi-type of analysis. As noted in the comments section, estimating the absorbed dose from the radiation activity level is never easy. My work here is very approximate, but does produce results in the same range as stated by the US National Institutes of Health. My overall objective is to build some tools to help me understand the effects that radiation in space and on Mars have on people.

Background

Reference Articles

My post was motivated by the following information I encountered on the web:

Source of Radiation in Tobacco

The EPA addresses the source of the radiation from tobacco:

Naturally-occurring radioactive minerals accumulate on the sticky surfaces of tobacco leaves as the plant grows, and these minerals remain on the leaves throughout the manufacturing process. Additionally, the use of the phosphate fertilizer Apatite – which contains radium-226, lead-210, and polonium-210 – also increases the amount of radiation in tobacco plants.

The radium-226 that accumulates on the tobacco leaves predominantly emits alpha and gamma radiation. The lead-210 and polonium-210 particles lodge in the smoker's lungs, where they accumulate for decades (lead-210 has a half-life of 22.3 years). The tar from tobacco builds up on the bronchioles and traps even more of these particles. Over time, these particles can damage the lungs and lead to lung cancer.

Figure 2 provides an excellent illustration of how polonium-210 (210Po), uranium-238 (238U), and lead-210 (210Pb) get into tobacco (Source: Mel Porter).

Figure 1: How Polonium Get Into Tobacco.

Figure 2: How Polonium Get Into Tobacco.

Appendix A goes into detail on how 210Po actually gets into the leaves because of 222Rn.

I found a number of quite different values quoted for the radiation level of tobacco leaves. I decided to choose the value that reflected the average radioactivity levels for US tobacco. US tobacco is more radioactive than others, possibly because of our use of slightly more radioactive fertilizers.

Radiation Exposure Definitions

Types of Radiation Doses

Absorbed dose
Energy absorbed by a kg of a substance. The absorbed dose is represented symbolically by DT,R, with T representing the specific tissue (e.g. brain) and R representing the specific type of radiation (e.g. x-ray). Absorbed dose is measured in units of Gray (Gy). By definition, 1 Gy = 1 joule/kg.
Equivalent dose
Equivalent dose is the absorbed dose weighted by the effect of the different types of radiation. The equivalent dose is represented symbolically by HT and computed by the formula {{H}_{T}}=\sum\limits_{R}{{{w}_{R}}\cdot {{D}_{T,R}}}, where wR represents the weighting for radiation effects relative to x-rays (wX-Rays=1). Equivalent dose is measured in units of Sieverts (Sv).
Effective dose
Effective dose is the equivalent dose weighted by the radiation sensitivities of the different tissues. The effective dose is represented symbolically by E and computed by the formula E=\sum\limits_{T}{{{w}_{T}}\cdot {{H}_{T}}}, where wT represents the weighting for tissue radiation sensitivity. The tissue radiation sensitivity is normalized so that weights for all tissues sum to 1. Effective dose is measured in units of Sieverts (Sv).

Radiation Units

Sievert
The Wikipedia defines the Sievert (symbol: Sv) as the SI derived unit of equivalent radiation dose. The Sievert represents a measure of the biological effect, and should not be used to express the unmodified absorbed dose of radiation energy, which is a physical quantity measured in Grays.
Gray
The Gray (symbol: Gy) is the SI derived unit of absorbed dose. Such energies are typically associated with ionizing radiation such as X-rays or gamma particles or with other nuclear particles. It is defined as the absorption of one joule of such energy by one kilogram of matter.

Analysis

My calculations use the same approach as David Gillies in his forum posting. However, my inputs ended up being different and I obtained a different result.

Discussion of Steady State Radiation Level

Over time, the radiation level emitted from cigarette smoking approaches a steady-state level. The steady state level is reached when the 210Po that decays each day is exactly cancelled by the amount of 210Po that is being inhaled every day. My analysis assumes that the smoker has reached steady state.

Definition of Units

Figure 3 shows the various units that I defined for this problem solution.

Figure 2: Units Defined for My Analysis.

Figure 3: Units Defined for My Analysis.

Wikipedia on Curie Unit Wikipedia on Sievert Unit Wikipedia on Gray Unit

Weighting of Effects By Radiation Type

Figure 4 shows the biological weighting factors for different kinds of radation. 210Po emits alpha particles, which have a weighting factor of 20 relative to x-rays.

FIgure 3: Weighting of the Different Radiation Types.

Figure 4: Weighting of the Different Radiation Types.

Polonium Characteristics

Figure 5 shows the relevant facts on 210Po.

Figure 4: Characteristics of Polonium-210.

Figure 5: Characteristics of Polonium-210.

Polonium characteristics

Radiation Model

Figure 6 shows my calculations for the effective radiation dose that a 1.5 pack a day smoker endures.

Figure 5: Radiation Calculations for 1.5 Pack a Day Smoker.

Figure 6: Radiation Calculations for 1.5 Pack a Day Smoker.

Radioactivity in Cigarette Smoke

Conclusion

I was looking for a simple example of computing the effects of radiation on a human. This example produces a result that is consistent with the data in the Wikipedia.

Appendix A: Source of Polonium

210Po is generated as a decay product from 222Rn. Here is the decay chain for 222Rn, which has 210Po as an intermediate product (Source). I highlighted the isotopes mentioned above.

  • 222Rn, 3.8 days, alpha decaying to...
  • 218Po, 3.10 minutes, alpha decaying to...
  • 214Pb, 26.8 minutes, beta decaying to...
  • 214Bi, 19.9 minutes, beta decaying to...
  • 214Po, 0.1643 ms, alpha decaying to...
  • 210Pb, which has a much longer half-life of 22.3 years, beta decaying to...
  • 210Bi, 5.013 days, beta decaying to...
  • 210Po, 138.376 days, alpha decaying to...
  • 206Pb, stable.

Save

Posted in Health | 4 Comments

Minnesota Winter Turns a Corner This Time of Year

Posted on 22-January-2014 by mathscinotes

We are going through a some very cold weather now in Minnesota. During late January, I start to daydream about warm weather. While daydreaming about going to my lake cabin today, I started to wonder when the average daily temperature in Minneapolis begins to increase (Minneapolis is the nearest large city to me). So I decided to go out to the Weather Underground and download the Minneapolis weather data since 1990, compute the daily averages, and smooth the data. I plotted this data in Figure 1. The minimum average daily temperature occurs on the 19th of January. So our daily average temperature is now increasing. That makes me feel like warm weather is not too far away.

Figure 1: Daily Average Minneapolis Temperatures During the Coldest Part of Winter.

Figure 1: Daily Average Minneapolis Temperatures During the Coldest Part of Winter.

Posted in General Science | Comments Off on Minnesota Winter Turns a Corner This Time of Year

Excellent Video on Breaking the Japanese JN-25 Code During WW2

Posted on 19-January-2014 by mathscinotes

Those of you World War 2 history buffs may find this video lecture on the breaking of the Japanese naval codes. I have read a number of books on the topic, but I did not know that so much work went into using machines to find patterns in the data.

Posted in History of Science and Technology, Military History | Tagged , , , | 1 Comment

Why Bother with Nitrogen in Tires?

Posted on 19-January-2014 by mathscinotes

Quote of the Day

Hope without a plan is denial.

— Time management expert

Introduction

Figure 1: Nitrogen has advantages for inflating tires..

Figure 1: Nitrogen has advantages tire inflation. (Source)

Every Friday afternoon, the hardware and software engineers sit down in our lunch room and chat about what occurred during the week. The discussion is always lively and includes management and engineers. It is my favorite time of the week at work. This week we discussed why automotive shops put "dry nitrogen" in car tires (Figure 1) – I am pretty sure all compressed nitrogen is dry because of the way it is processed.

A couple of our engineers are motorheads and they use nitrogen to fill the tires of their personal vehicles. This post is an extension of my earlier post on the variation of tire pressure with temperature.

We normally fill our tires with pressurized air, which is 78% nitrogen. There are four main reasons why professional car people use 100% nitrogen:

  • Nitrogen-filled tires retain their pressure longer.

    All tires eventually lose air pressure with time as the gas diffuses through the tire. Nitrogen has a lower rate of diffusion from tires than oxygen, so the tire retains it pressure longer. The permeability of a gas through a material is a function of the gas and the material. For example, the permeabilities of nitrogen and oxygen through poly-isoprene are (Source)

    • Oxygen: {{\kappa }_{{{{O}_{2}}}}}=4.6\cdot {{10}^{{-13}}}\frac{{\text{c}{{\text{m}}^{3}}\left( {\text{STP}} \right)\cdot \text{cm}}}{{\text{c}{{\text{m}}^{2}}\cdot \text{s}\cdot \text{Pa}}}

    • Nitrogen: {{\kappa }_{{{{N}_{2}}}}}=1.6\cdot {{10}^{{-13}}}\frac{{\text{c}{{\text{m}}^{3}}\left( {\text{STP}} \right)\cdot \text{cm}}}{{\text{c}{{\text{m}}^{2}}\cdot \text{s}\cdot \text{Pa}}}

    Where STP standpoints for standard temperature and pressure. As you can see, the permeability of oxygen through synthetic rubber is substantially greater than that of nitrogen.

  • Nitrogen gas is dry -- it has virtually no water in it.

    We isolate nitrogen from the air industrially by the fractional distillation of liquified air. This process removes all water from gas. Using a dry gas provides two advantages:

    • Removing water eliminates a source of corrosion.

    • Removing water results in less pressure variation with temperature. See Figure 2 for an example (Source).

      Figure 1: Variation of Tire Pressure with Moist Air.

      Figure 2: Variation of Tire Pressure with Moist Air.

  • Unlike oxygen, nitrogen gas does not promote corrosion.

    At least for iron compounds, oxygen promotes rust. Oxygen and water together are bad for iron-based materials.

  • Nitrogen is relatively cheap.

    There are lots of dry gasses available. Nitrogen is about as cheap as you can get.

After we completed our discussion of tire pressure, we then discussed how we need to liven up our discussions. Sitting around on a Friday afternoon discussing tire pressure sounds really boring. I guess that's what life is like with a bunch of engineers around.

When I worked on torpedoes, we used to backfill them with nitrogen. The main concern there was corrosion.

Posted in General Science | Tagged , , , | 1 Comment

Very Cool Wind Maps

Posted on 16-January-2014 by mathscinotes

US Wind Map

US Wind Map

World Windmap

World Windmap

Posted in General Science | Comments Off on Very Cool Wind Maps

Radioactive Banana Math

Posted on 16-January-2014 by mathscinotes

Quote of the Day

I never did a day's work in my life. It was all fun.

— Thomas Edison

Introduction

I have been reading about the hazards of space travel to Mars. During this reading, I occasionally see references to space radiation hazards in terms of Banana Equivalent Dose. I find this a strange unit. Then today I read a blog post by Anne Marie Helmenstine that discussed how bananas are slightly radioactive. I liked her discussion and I thought I would go through the math here.

Background

Why are Bananas Radioactive?

Bananas are radioactive because they contain potassium and potassium has an isotope (40K) that is radioactive. This same isotope is present in humans, which means that humans are also slightly radioactive.

Banana Data

Figure 2 shows the data that Google puts out when you type in "Amount of Potassium in a Banana".

Figure 2: Banana Characteristics from Google Search.

Figure 2: Banana Characteristics from Google Search.

While there is a lot of data here, I will only use the amount of potassium (422 mg) in a typical banana (118 gm) to estimate the rate of radioactive decay.

Measuring Banana Radiation

Here is a Youtube video of a person measuring the radiation from a banana. You can hear the sound of the Geiger counter. Ignore his comments about bananas being dangerous because of radiation levels. We are always exposed to low-level radiation.

Analysis

Figure 3 shows my quick analysis of Anne's results.

Figure 3: My Analysis of the Radiation Emission from a Banana.

Figure 3: My Analysis of the Radiation Emission from a Banana.

Definition of Mole Information on Potassium

Conclusion

Numbers all confirmed. I also read that Brazil nuts are relatively radioactive.

Posted in General Science | 2 Comments

Funny the Little Things that Children Remember ..

Posted on 14-January-2014 by mathscinotes

Quote of the Day

A startup is a temporary organization searching for a repeatable and scalable business model.

- Steve Blank

Gamera on MST3K

Gamera on MST3K

My wife and I went out to dinner this weekend with our oldest son and his girlfriend. We had a good time reminiscing about "the good old days". I always find it interesting that both of our sons fondly mention the little things in their childhoods as being special moments. This weekend, my oldest son recalled how we would spend Saturday morning watching Mystery Science Theater 3000 (MST3K) . All three of us would be laughing the whole time. There is no sound as wonderful as that of children laughing. We still talk about how fun it was to watch Japanese monster movies on MST3K. As far as I am concerned, that is the only way to enjoy Godzilla and Gamera.

Godzilla on MST3K

Godzilla on MST3K

In addition to our Saturday morning ritual, we also had our little rituals during the week. I am not really a video game person (unless you count Silent Hunter), but I would play games with them like Contra and Dizzy for a few minutes every day. At that time, I would never have suspected that I was creating such fond memories with my sons. My youngest son has said that my gaming with him were his fondest memories.

That filled my heart with joy. Who knew that something as simple as playing video games would provide my son with memories that would last a lifetime? It really was great to hear. There is so much you can do with video games nowadays as they adapt to what the gamer, like my son, needs, with game trainers by FLiNG that can help them navigate their way through without missing out on all the fun, people are loving the new changes. Although I wasn't much of a gamer myself, I really did find myself having a lot of great fun getting stuck into the gaming world, and immersing myself as one of the many characters that you can choose from. I specifically remember my friend telling us to try Elder Scrolls Online if we wanted to play something that was out of this world. Before doing so though, he told me to look into this best build eso to see how you can take down a group of mods while leveling, as well as other advantages too. Well, I looked into this, and this is something that I definitely considered at the time but didn't follow through with. As I'm reminiscing on these times, I might get my son and play a game with him now. That would be fun.

My favorite memory occurred late one night when my sons snuck out of their bedrooms and came downstairs to watch TV with me. It was long after their bedtime, but I decided to let them stay up late. They loved to stay up late, but I knew they needed their sleep. One night I was watching the movie To Kill a Mockingbird and I decided to let them watch it with me. They were mesmerized by it -- I think they really identified with Scout. We had a wonderful discussion about people, discrimination, and how important it is to treat everyone with respect.

I sometimes hear folks say that television and video games are problems for children. Used properly, television and video games can also provide wonderful memories.

Posted in Personal | 1 Comment