Thank You for using JBallistics. I are sure that you will find this program both useful and instructive.

Although JBallistics does have its limitations (described in a separate section) the accuracy of the model is very good and can be used with confidence in assessing the absolute performance of a particular cartridge as well as its performance relative to other cartridges.

You should have no trouble using the buttons and menu bars to navigate through the various functions. The text fields are accessible with a mouse click. Data into these data fields is automatically checked for validity. That means that numeric data fields will only allow: numbers, only one decimal point, and a minus sign in the leftmost position only. Most windows with many text fields recognize the arrow keys to facilitate moving the focus from one field to the next.

JBallistics plot functions include a number of features to enhance the display and comprehension of that data. Select the 'File' menu to print the plot. Select the 'Color' menu to change the background color of the plot. Select the 'Resize' menu to enlarge or reduce the size of the plot. Select the 'Replot' menu to redraw the plot using only data contained in a box drawn on the plot by dragging the mouse while holding down the left mouse button. You can 'zoom' in as many times as you like. The original plot can be retrieved at any time. Select the 'Points' menu to toggle the display of the actual data points used to construct the plot. The 'Labels' menu is used to show the plot labels in a separate window. When many data lines appear on the same plot, the lines are sometimes hard to identify. This separate window makes it easier to identify them. The mouse arrow when passed over a plot acts as a digitizer showing the position of the mouse in data units. You can place the mouse over a position on the curve and instantly read the (x,y) coordinate values displayed in the frame title bar. You may have to drag the right side of the plot over a bit to be able to see this part of the header.

JBallistics is written as a Java Application. It should run equally well on any computer that has a compatible Java Virtual Machine, an interpretor that executes the Java bytecodes for the computer and operating system you are using. That includes Windows 95/98/NT, Mac OS, Linux, Solaris, etc. JBallistics requires the Java 2 version.

Since Java is a multithreaded language, you can have things happening simultaneously all over the screen, limited only by available memory and processing power. Because JBallistics displays much information on the screen at the same time, it was designed to require a minimum screen resolution of 1024x768 pixels. You can also choose to leave some windows up on the screen after computations are completed and perform other tasks to perform comparisons.

JBallistics has been developed to be useful at various levels of complexity depending on your requirements. Three ballistics models are available:

These are in order of increased complexity for the user to be able to extract useful results.

The Vacuum Trajectory is of limited usefulness since its primary model is that of gravity and neglects atmospheric effects. It is included here in the event the user wishes to compare the results with the other models to assess the impact of aerodynamics on ballistic trajectories.

The Point Mass Trajectory will be the primary model the user will want to use. It includes an advanced gravity model, aerodynamic effects, Coriolis effects, spherical Earth corrections, etc. This model uses the concept of standard drag functions and ballistic coefficients. Drag functions relate the velocity of the bullet to the aerodynamic drag at that velocity. The ballistic coefficient describes how a particular bullet performs relative to a standard projectile used to determine the standard drag function. The most prominent assumption made for the Point Mass Model is that it neglects the additional aerodynamic effects of bullet yaw. This yaw contributes to significant bullet side drift at longer ranges.

The Modified Point Mass Model does consider the effects of yaw by introducing additional terms to the equations of motion. The advantage here is that, at longer ranges, the real-world side drift of the trajectory will be predicted. The disadvantage is that an accurate simulation of this side drift requires knowledge of additional aerodynamic coefficients and information about the bullet. JBallistics incorporates advanced models which predicts these coefficients based on the dimensions of the bullet which you must provide. The program provides tools for you to make this process simpler. For more sophisticated users with access to actual experimentally measured values for these aerodynamic coefficients, tabular data entry features are provided.

Please read the Ballistics Primer section for more detailed information on each of these models.

The field of ballistics is divided into three areas: interior ballistics which deals with the behavior of the bullet inside the barrel, exterior ballistics which concentrates on atmospheric flight, and terminal ballistics which is concerned with the impact of the bullet on its target. This program deals with exterior ballistics.

Bullets do not fly in a straight line. Rather they fly in an arc which begins to drop off immediately after leaving the barrel. Bullets also drift naturally to one side. The further the distance from the barrel, the greater the drop and the greater the drift. The science of Ballistics attempts to quantify these values. When you zero the sights on your rifle you are correcting for these effects. For a 200 yard zero, for example, the barrel actually point upwards slightly above the target and probably a bit to the left of the barrel centerline. The arc's path carries the shot to the desired point of aim.

Gravity and the atmosphere are the two major elements that contribute to the techniques used to predict a trajectory. Gravitational effects are fairly simple to model, atmospheric effects are not. The biggest problem is that the retarding effects of the air on a projectile in motion is too complex a system to model with complete accuracy. The approach then is to develop approximations that work well enough to solve problems with sufficient accuracy for certain practical applications. It is then important to realize the limitations of the techniques employed.

The approach taken since about the beginning of the twentieth century has been to test fire a number of projectiles of different shapes and then develop a set of functions that describe the atmospheric retardation effects on these projectiles as a function of their velocity relative to the speed of sound (mach number). These are the so called 'G' functions probably named for the Gavre Commission of the French Naval Artillery which performed the early tests starting in 1873. In 1900 Colonel James M. Ingalls of the U.S. Artillery published English units version of data analyzed by Russian General Mayevski from test firings conducted by the German Krupp Factory from 1875 to 1881. The Mayevski projectile was similar to the early Gavre projectile and hence produced very simlar results. Test firings for various shaped projectiles continued until about 1940. After World War II the U.S Army Ballistic Research Laboratories conducted experiments using advanced techniques including supersonic wind tunnels and radar ranges and thus refined earlier test results as well as producing data for a host of new projectiles. These results form the basis for the classic drag functions known today as G1, G2, G5, G6, G7, and G8 for the projectile type they describe. The Type 3 projectile was essentially a check on the firing data for the G1 projectile and the Type 4 projectile was of an unusual shape that has little interest today. A GS function exists as applied to spherical shaped projectiles of limited size.

To compute a trajectory for a projectile of different shape from these standard projectiles, the concept of a Form Factor is introduced:

Form Factor = Cd of Bullet / Cd of Standard Bullet

Cd = Coefficient of Drag from Drag Function

To incorporate the effects of a different size and weight projectile from the standard, the concept of Sectional Density is introduced:

Sectional Density = Bullet Weight / [(Diameter)(Diameter)]

To combine these values to account for shape variation as well as size and weight, the concept of a Ballistic Coefficient is introduced.

Ballistic Coefficient = Sectional Density / Form Factor

The standard projectiles therefore have a Ballistic Coefficient of 1. for their respective drag functions. Thus, with knowledge of a standardized drag function and a ballistic coefficient, it is possible to produce a predicted trajectory which is pretty close to what will actually happen when the projectile is fired. This technique forms the basis for the Point Mass Algorithm that can be used in JBallistics to predict a trajectory.

Today, most manufacturers of small arms ammunition have settled on the G1 drag function as the standard and publish ballistic coefficients for their bullets relative to that. Unfortunately, the G1 projectile with its flat base and blunt nose is not really representative of most modern bullets with boattail bases and sharper noses. Also, as you can tell from the equations above, the ballistic coefficient should really vary with mach number just like the drag function. A bullet with a different shape from the standard bullet to which it is being compared does not have a drag function that matches the shape of the standard drag function at every point. Published ballistic coefficients are therefore only valid at one average velocity. In spite of this, the G1 model produces results quite good at least up to about 500 yards.

If one has the proper data, for contemporary rifle bullets with longer noses, the G7 drag function is best for boattailed bullets and the G8 drag function is preferred for flat base bullets.

As an example, for the boattailed 7.62 mm M80 bullet, the actual G1 ballistic coefficient varies in a range from about 0.25 to 0.40 from mach 1 to about mach 3. The G7 ballistic coefficient varies only from about 1.9 to 2.1 in the same mach number range. Hence, the G7 drag function is superior in this case. A good average G7 ballistic coefficient for this bullet is 0.198 out to about 1000 yards.

The advent of modern digital computers made possible the numerical solution of the actual differential equations of projectile motion. This has permitted advanced modelling of trajectories including yaw effects. Experimental techniques such as spark photography and hypersonic wind tunnels has made possible the determining of the aerodynamic coefficients required as inputs to these equations. JBallistics can use a Modified Point Mass Model which models bullet drift due to yaw effects.

In 1969, R.H. Whyte of the General Electric Company developed a computer program called SPINNER which correlated large amounts of research data to produce reasonably accurate predictions of the aerodynamic coefficients of projectiles. This model was updated in 1973 and again in 1979. The 1979 algorithm update to this code is incorporated in JBallistics as a part of the Modified Point Mass Model. R.H. Whyte now offers a very advanced ballistics program called PRODAS containing further refinements, enhancements, and capabilities.

In 1974, R.L. McCoy of the U.S. Army Ballistics Research Laboratories developed a computer program called McDrag based on the use of aerodynamic similarity laws to correlate a large volume of drag coefficient data to predict drag functions for projectiles. The inputs to McDrag are the size parameters of a bullet. The algorithms in McDrag are also incorporated into JBallistics and form the basis for determining the Custom Drag Functions. McCoy also developed a program called INTLIFT based on M.A. Morris's RARDLIFT code which predicts additional aerodynamic coefficients. A version of this with corrections by Brad Millard is also incorporated in JBallistics for use by the Modified Point Mass Model. The gyroscopic stability analysis presented by the Modified Point Mass Model in JBallistics is based on McCoy's 1986 McGyro program.

The section on 'How To Use JBallistics' will discuss how to use the various algorithms and calculation functions available in the program.

To know more about external ballistics, please consult these references:

Books:

Technical Reports:

On the Internet:

There are many levels of complexity in JBallistics depending on your requirements, interests, and availability of data. This section will describe how to perform some basic functions from simple to more complex.

SIMPLE:

Start by defining some of the Shooting Parameters to meet your needs. Use the default values for these Shooting Parameters except the following:

Select a bullet and double click on it (or single click on it then click on the 'Work With Selected Cartridge Data' Button.) The 'Work With Cartridge data' window should come up.

Accept the stored data provided, choose an analysis such as 'Bullet Drop' then click on that button. View the plot that comes up. You will also see the computed muzzle angle required for the selected Sight Zero Distance displayed in that text field.

As an alternative, after defining your Shooting Parameters, select a few bullets in the bullet list with single clicks while holding down the Control Key. Then click on one of the Plot buttons above the bullet list to produce a plot comparing the trajectories of the selected bullets.

A BIT MORE COMPLICATED:

Determine which drag function is more suitable for your bullet than the G1 function selected as the default. As stated in the Short Ballistics Primer, the G7 drag function is best for boattailed bullets and the G8 drag function is preferred for flat base bullets. Bring up the 'Work With Cartridge Data' window for that bullet, Click on the Ballistic Coefficient text field in that window, blank out the field with the Del key, then click on the button for the plot you want to see. JBallistics will compute the Ballistic Coefficient for the selected Drag Function (by assuming the Form Factor = 1.0) and display it in the text field area.

MORE COMPLICATED WITH CHRONOGRAPH DATA:

Choose the desired Drag Function for your bullet in the Shooting Parameters window. Select 'Calculations' from the Main menu and bring up a 'Ballistics Calculations' window. Click on the 'Ballistic Coefficient 1' tab (it should already be visible as the first tab). Enter your velocity, distance, and/or time data. Click on the 'Calculate' button to calculate the Ballistic Coefficient. Enter that Ballistic Coefficient in the 'Work With Cartridge Data' window to calculate the trajectory.

MORE COMPLICATED WITHOUT CHRONOGRAPH DATA:

Choose the desired Drag Function for your bullet in the Shooting Parameters Window. Select 'Calculations' from the Main menu and bring up a 'Ballistics Calculations' window. Click on the 'Form Factor 2' window. Choose the Drag Function in that window, select the Ogive Type and Base Type for the bullet, enter the bullet diameter and nose length, then click on the 'Calculate' button to compute a Form Factor. Next, click on the 'Ballistic Coefficient 2' tab, enter the bullet diameter, weight, and the computed Form Factor, click on the 'Calculate' button to compute a Ballistic Coefficient. Make sure the desired Drag Function is selected in the Shooting Parameters window. Use the calculated Ballistic Coefficient in the 'Work With Cartridge Data' window to calculate the trajectory.

EVEN MORE COMPLICATED WITH CHRONOGRAPH DATA:

Here you will proceed similarly to 'MORE COMPLICATED WITH CHRONOGRAPH DATA' but you will define a Custom Drag Function for use with the Point Mass Model. In the Shooting Parameters Menu under 'Drag Functions' select the menu item 'Show Custom Drag Function' window. One of the supplied Custom Drag Functions will be displayed. You may choose one of these or add your own. Let's add our own here for this description. Click on the 'Name Of Drag Function' text field and overwrite the text with your new bullet name. Enter the size parameters for your bullet in the bullet data fields. Refer to the sections on Drag Functions for details and more information on this. You may also click on the 'Display Bullet Shape Graphics' button to bring up that tool to help you determine these size parameters. See the section on that topic for more information on how to use this. Save your new Custom Drag Function using the 'Copy to New Function Name' menu item in the File Menu in the Custom Drag Function window. Your new drag function should also be automatically the one selected in the Shooting Parameters window. You may click on the 'Compute and Plot Drag Function' button to view your new drag function. Make sure you have entered your Ballistic Coefficient in the proper text field and click on the button that corresponds to the trajectory analysis you want to see.

EVEN MORE COMPLICATED WITHOUT CHRONOGRAPH DATA:

Here proceed as immediately above except that you will blank out the Ballistic Coefficient field in the 'Work With Cartridge Data' window and let JBallistics calculate and display the Ballistic Coefficient.

MOST COMPLICATED:

The most complicated analysis involves specifying the 'Modified Point Mass' Model in the Shooting Parameters window. You may specify any drag function you wish here just be sure any Ballistic Coefficient you specify is appropriate for that drag function. Alternatively, blank out the field and let JBallistics calculate it (again assuming the Form Factor = 1.) A further complication here is that you have the option of either letting the program calculate the required aerodynamic coefficients or specifying them yourself in the table provided (provided you know what they are.) Under the 'Aero Coef.' menu in the Custom Drag Function window, select either the 'Use Calculated Coefficients' or 'Use Tabular Coefficients' menu item. Selecting 'Use Calculated Coefficients' is simpler, no further work is required. To enter tabular coefficients, select the 'Show Coefficient Table Window' menu item. A set of tabular coefficients may be associated and stored for each Custom Drag Function you have. When you select the menu item, the tabular coefficient set for the currently displayed Custom Drag Function is shown. Initially the table is blank, you must fill it in the best you can. Blank entries are ignored. Use the arrow keys to facilitate moving the cursor around the table. See the section describing this table for more detailed information. You should now be ready to calculate the trajectory. Remember with these coefficients, the quality of the computed trajectory is dependent on the quality if the data supplied or, in other words, Garbage In, Garbage Out. Be sure your data source for all these coefficients is reliable. After a Modified Point Mass trajectory has been calculated you may view additional plots by accessing the 'Last Analysis Plots' menu in the Custom Drag Function window.

JBallistics is supplied with data for 1500 rifle and handgun cartridges. More cartridges can be added to the data base as you wish. JBallistics users will want to evaluate cartridges based on their intended use.

In addition, JBallistics can be used as an educational tool to understand how bullet performance varies with changes in various shooting specifications. For example, JBallistics allows you to:

All the research that I have done on Ballistics has left me with the impression that ballistics is every bit as much art as science. The primary difficulty is the unpredictable effects of the air on the trajectory of the projectile. There is no way to predict the precise effects of the complex fluid medium known as the atmosphere on the flight of a bullet at every point along its path. We can do very well but we can't be perfect. The unpredictable variations in wind, temperature, pressure, etc. make atmosphere modeling a statistical science not an exact one. The best we can do is to assume an ideal atmosphere and rely on the law of averages. As the trajectory distance increases and as the angle of launch increases the errors become larger.

JBallistics incorporates three models from which the user may choose. These range from fairly simple to rather complex. Increases in complexity can potentialy yield better results but then the models are vulnerable to the quality of the data that drives them.

The simplest model is the vacuum trajectory which neglects the effects of aerodynamics. You can probably guess that this model makes predictions that diverge widley from actual results. Nevertheless, this model is still used today to predict the trajectory of high-arch, low speed mortar fire. It is this low speed where atmospheric effects are smaller that makes this possible. Other than very low speed fire, this model is of limited use other than for instructional purposes.

The most useful model is the Point Mass Trajectory. This model includes atmospheric effects but is limited by the necessary assumption of an ideal atmosphere. Correction factors are included for various atmospheric parameters but no atmosphere is ideal. Correction factors are even included for variations in these parameters to account for additional altitude of the bullet along its trajectory. However, as previously stated, the precise atmospheric effects cannot be predicted or even known for every condition.

Further, the Point Mass Model depends on the use of generalized drag functions and ballistic coefficients to describe how a particular bullet performs relative to a standard drag function. The limitation here is that every bullet should have its own drag function and not matched to a limited selection of generalized shapes. Moreover, the ballistic coefficient is a single quantity, but should, in fact, itself be a function which varies with mach number as does the drag function. JBallistic allows you to define your own custom drag functions and the limitation then becomes how accurate that function is at each point.

The Point Mass Model also neglects the effects of bullet yaw and trajectory drift and oscillations that occur in actual flight.

The current version of JBallistics does not allow you to produce comparison plots of bullets with each bullet using a different model or custom drag function. The model selected applies to all bullets in the comparison plot.

JBallistics is provided with ballistic coefficients for each bullet in the database. These are supplied by the bullet manufacturers and all assume the G1 drag function shape. If you examine the form of a G1 projectile, you will see that it doesn't apply to the modern boattail bullet shapes fired for accuracy today. The G7 or G8 shape is much more appropriate but general availablity of these ballistic coefficients is almost nonexistent.

Also, the muzzle velocity values provided are for a known barrel length. Muzzle velocities at other barrel length must be estimated with an approximation formula. It is unknown how accurately this approximation works.

It is estimated that the accuracy of the Point Mass Model is limited to the downrange distance where the bullet velocity has decayed to about one half of its muzzle velocity. Up to 500 yards the results are quite usable.

The Modified Point Mass algorithm attempts to correct for the effects of yaw drag on the bullet and produce realistic results for the side drift. This is dependent on the accuracy of the predicted yaw magnitude at each point along the trajectory. This depends on the accuracy of the aerodynamic coefficients predicted by JBallistics based on the bullet dimensions. These aerodynamic coefficient prediction models are based on statistically averaged results of measurements of a large number of projectiles. While they are quite good, they are not completely accurate for every projectile. The aerodynamic coefficients are accurate to within a standard deviation of between 10 to 25 percent. Further, JBallistics make estimates of the center of gravity location as well as the Axial and Transverse Moments of Inertia based on bullet dimensions. These are just estimates and are not completely precise.

The overall accuracy of the Modified Point Mass model, in general, will be better than for the Point Mass Model. If you have experimentally verified aerodynamic coefficients, the results may be good up to a downrange distance of a few thousand yards. The results for the calculated coefficients somewhat less than that. The limitation of the Modified Point Mass model is limited to a lauch angle of approximately 70 degress. Beyond that, accuracy will be substantially reduced. Many bullets will "fail to trail" if launched at a steep angle. That is, they will rise nose first but fall base first. This behaviour is beyond the ability of the Modified Point Mass model to predict.

A better trajectory model would be a six-degree of freedom algorithm (6-DOF). This would require the precise knowledge of trajectory initial conditions as well as a larger variety of aerodynamic coefficients. This makes use of such a model impractical for all but ballistics laboratories concentrating on research on the behaviour of a particular projectile.

To select a cartridge from the list, simply place the mouse arrow over the cartridge name and click the left mouse button. To select an additional cartridge, place the mouse arrow over that cartridge name, hold down the Ctrl key and click the left mouse button again. Two cartridges are now selected. You may select as many cartridges as you wish in this manner. To select a range of cartridges, select one cartridge then place the mouse arrow over a second cartridge name, hold down the Shift key and click the left mouse button. The entire range of cartridges between the two will be selected. You may select various cartridges from the database using combinations of these techniques.

Once you have selected a cartridge(s) you may click on the 'Work With Selected Cartridge Data' button to bring up all the data windows for the selected cartridge(s).

You may select a single cartridge and immediately bring up the data window for that cartridge by double clicking on its name in the list.

After you have 'selected' the cartridge from the list by clicking at least once on the list area you can shortcut immediately to cartridges beginning with a number by pressing a number key. If you press the '3' key, for example, cartridges beginning with '3' will appear at the top of the list.

You may clear all selections by clicking on the 'Clear All Selections' Button below the list of cartridges.

The data file maintenance functions are accessed from the File menu on the main task bar.

REFORM DATA FILE:

This function allows you to 'fix' any structural errors in the data file that may occur as a result of power interruptions during critical data update procedures or due to disk media problems. The window shows the current capacity of the data file and permits you to overwrite this field with a number of your choice.

The current capacity of the file is the number of data records that can be contained before the automatic self-expansion procedure occurs when the file becomes full. If you wish to expand or contract the capacity of the file, enter a new number. Click on 'OK' to proceed with the reform process or 'Cancel' to exit. If you enter a different value for the file capacity, that number will be adjusted by JBallistics to verify that there is enough space to hold the current number of entries and will be increased to the closest 'prime number' greater than or equal to the value you entered.

ADD DATA FROM CSV FILE:

This function allows you to enter data to the database in batch form using an external text mode file as the data source. The data file uses two different record types using the comma separated values (CSV) format.

Record Type 1:

Cartridge Name, Barrel Length, Barrel Twist Rate

Barrel Length is for the muzzle velocites of the bullets in Type 2 records. The Barrel Twist Rate is not currently used.

Record Type 2:

Diameter, Weight, Ballistic Coefficient, Muzzle Velocity, Name

Bullet diameter is in inches, weight is in grains, muzzle velocity is in feet/second.

Here is an example of data for five Remington .223 bullets followed by three Winchester .308 bullets:

223 REMINGTON,24.,9.
.224,40.,.218,3800.,HORNADY V-MAX MOLY (83253)
.224,40.,.202,3700.,WINCHESTER BALLISTIC SILVERTIP (SBST223A)
.224,50.,.204,3400.,AMERICAN EAGLE (FEDERAL) JHP (AE223G)
.224,50.,.238,3410.,WINCHESTER BALLISTIC SILVERTIP (SBST223)
.224,50.,.238,3300.,BLACK HILLS V-MAX (M223N7)
308 WINCHESTER (7.62MM),24.,12.
.308,123.,.274,2936.,LAPUA FMJ (4317527)
.308,125.,.316,3028.,HANSEN POSU-FEED SP (HCC308E)
.308,146.,.441,2627.,NORMA FULL JACKET (17662)

JBallistics uses RAM memory as it is needed up to that amount available on your computer up to 256 MB. If you have more memory than that you can alter the startup command in the shortcut used to start the program (right click on the JBallistics startup icon with the little arrow, not the .ico file). The -Xmx256m parameter defines that maximum. Change the 256 to whatever value is appropriate. If you experience sluggish performance or nonfunctionality of some buttons, it is possible that you have consumed all available memory. JBallistics will begin using virtual memory (hard disk) after the maximum has been reached. Performance at that point will be drastically reduced.

You can check this by selecting Help - About from the main menu bar. A system information window will appear showing, among other things, memory utilization information. You can click the 'Garbage Collection' button to try to free up some memory although this will normally be done automatically by the program. In any event, you can see if you have a memory consumption problem. Closing open windows will release memory. Screen graphics use alot of memory. This includes the background images on the main screen which you can disable/reenable at any time.

Screen resolution plays a large part in memory consumption. At 1024x768 a full-screen plot uses nearly 3.2MB assuming 32-bit true color. At 1600x1200 that same image consumes about 7.7MB. With the buffering techniques used to generate the plots, these values actually go quite a bit higher.

You may work in either English or Metric units. The units mode is selected from the main menu bar. Changing the mode from English to Metric while windows are open on-screen will change the units on the fly except for existing displayed data plots.

The calculator is available from the main menu bar. To use the calculator, enter a mathematical expression on the working line then click 'Calc'. The numerical answer will be placed in the R register. To perform another calculation, enter another expression, click on 'calc', and that answer will be placed in the R register while the previous contents of the R register will be moved to the X register. Any contents of the X register will move to the Y register, the Y will move to the Z register, the Z will move to the T, and the previous contents of the T register will be lost.

The following symbols are valid for use on the "Expression" line:

All exponentiations will be done first, followed by multiplication and division, then addition and subtraction. This order of precedence may be altered by using parentheses to specify higher priority operations. Further, the contents of the r,x,y,z, and t registers may be used in the expression. The following are examples of valid expressions:

The results will be maintained to approximately 16 digits of precision. Division by zero is not permitted.

You can use either the buttons on the keypad to enter information or use the keyboard directly. The 'Clear' key erases pending expressions on the working line.

You may also use the menu bar Copy, Cut, and Paste clipboard operations to transfer results to the car data entry screens. Do this in the same way as any other program.

This window provides a number of ballistics related calculations accessible through the tabbed panes. Click on the tab to access the fields for that computation. Each calculation is described in its own help window. Note that you may have more than one of these windows on the screen at the same time by repeatedly accessing the menu item on the main menu bar. The data entry fields have a white background, the results fields have a grey background and are not accessible.

Here, you enter a known velocity for your bullet, and then either a second velocity or a time value followed by the distance travelled by the bullet for that second velocity or time value. Use the combo box to tell the program which value you wish to use. When you click on the calculate button, the other value (velocity or time) will be calculated and displayed. The resultant Ballistic Coefficient will be that coefficient for the drag function selected in the Shooting Parameters window. For example, if G1 is the currently selected drag function, enter 2600 ft./sec. for Velocity 1, 2500 ft./sec. for Velocity 2, leave time blank, Specify 'Use Velocity 2' for Select, and 20 yards for distance. Click on the calculate button and a G1 Ballistic Coefficient of 0.178 will be displayed. You will also see that the time to travel those 20 yards is 0.02353 seconds. The computation also incorporates the weather parameters currently defined in the Shooting Parameters window. The formula for this is as follows:

bc = [Cd(mach)](pi)(rho)(Vavg)(Vavg) / [(8)(a)]

where,

bc	=	ballistic coefficient
Cd	=	drag function
mach	=	mach number (bullet velocity / speed of sound)
rho	=	atmospheric density
Vavg	=	geometric average bullet velocity, SQRT(V1*V2)
a	=	bullet acceleration over the interval

if you know V2 then:

time = distance / Vavg

if you know time then,

V2=(distance / time)^2 / V1

and,

a = (v1-V2) / time

Note that the speed of sound as well as atmospheric density is a function of weather parameters.

This calculation computes the ballistic coefficient when the bullet diameter, bullet weight, and form factor are known. Some early ballistics techniques allowed the determination of a form factor from charts where you could match the shape of your bullet to known shapes and extract a form factor for a particular drag function (see Hatcher's Notebook). The formula used here is:

bc= weight / [(form factor)(diameter)(diameter)]

This function calculates the kinetic energy of the bullet. Enter the bullet weight and the velocity in the appropriate units, click on the button, and the program will display the kinetic energy. The formula used is:

Kinetic Energy = (Mass)(Velocity)(Velocity) / 2

where,

bullet Mass = weight / (acceleration due to gravity)

This function calculates the momentum of the bullet. Enter the bullet weight and the velocity in the appropriate units, click on the button, and the program will display the momentum. The formula used is:

Momentum = (Mass)(Velocity)

where,

bullet Mass = weight / (acceleration due to gravity)

This function calculates the recoil velocity and energy of a recoiling firearm. Enter the bullet, powder, and gun weight, muzzle velocity, and powder type. The recoil velocity and energy are calculated as follows:

rv = (ri)(g) / (gun weight)

re = (gun weight)(rv)(rv) / (2)(g)

where,

rv	=	recoil velocity
re	=	recoil energy
ri	=	recoil impulse
g	=	acceleration due to gravity

and,

ri = (bullet weight)(velocity) + [(K)(powder weight) / g]

where,

K = Powder gas escape velocity constant, (for smokeless powder = 4000) (for blackpowder = 2000)

The form factor of a bullet is computed using the bullet's diameter, weight, and ballistic coefficient:

form factor = weight / [(ballistic coefficient)(diameter)(diameter)]

The form factor is a dimensionless quantity.

This function computes the form factor of a bullet using the bullet diameter, nose length, bullet base type, and bullet ogive type. Specify the desired drag function for which you want to compute the form factor, click on the compute button, and the form factor will be computed. The G2 drag function is not available here as a selection.

A tangent ogive nose is one where the bullet's ogive is tangent to the bullet's body where the nose and the body meet. A secant ogive nose is one where the center of the circle that describes the ogive is centered behind the point where the nose and body meet. In the Bullet Shape Graphics Tools function, this is known as Ogive Center Offset. A tangent nose has a zero Ogive Center Offset. A secant nose has a non-zero Ogive Center Offset.

Some rules of thumb for using this function are:

For flat-based bullets where the nose length to diameter ratio is between 1.0 and 1.7, specify the G1 drag function. For conical nose bullets, specify the tangent ogive shape.

If the ratio is greater than 1.7 specify the G8 drag function. Another possibility for tangent ogives is to specify the G6 drag function but G8 is usually better. G7 can be used for boattailed bullets where the ratio is greater than 1.7 specify the G7 drag function. The G5 function is an alternative but not usually as good as the G7.

For the sphere drag function, sphere diameters from 0.05 up to 0.7 inches are valid.

There is no specific algorithm here. The program uses table lookup and interpolation to obtain its result. The basis of the data is the work performed by H.P. Hitchcock and presented in various Ballistics Research Laboratories (BRL) reports from 1939 to 1952.

Enter the sight Radius (the distance between the front and rear sights), the aimpoint error (positive for an error to the high side, negative for an error to the low side), and the target distance. The program will calculate the required sight adjustment to compensate for the error.

To raise the point of impact:

LOWER the FRONT sight or RAISE the REAR sight.

To lower the point of impact:

RAISE the FRONT sight or LOWER the REAR sight.

To adjust the rear sights, the formula is:

Rear Sight Adjustment = (sr)(ae) / (dt)

To adjust the front sight, the formula is:

Front Sight Adjustment = (sr)(ae) / (sr + dt)

where,

sr	=	sight radius
ae	=	aimpoint error
dt	=	distance to target measured from the front sight

Because dt is always much, much greater than sr, the two formulas yield essentially the same result. JBallistics presents the average of the two results.

The computation of angle correction is more complicated:

Rear Sight Angle Correction = asin[(ae + hr) / (sr + dt)] - asin[hr / (sr + dt)]

Front Sight Angle Correction = asin[(ae + hf) / dt] - asin[hf / dt]

where,

hr = (ae)(-sr - dt) / (rsh - sr - dt)

hf = (-ae)(dt) / (fsh - dt)

rsh = rear sight height

fsh = front sight height

In practice, the specification of front and rear sight height has no appreciable effect on the result (within 3 significant figures) for any reasonable values (2.5 in. rear and 1.5 in front is assumed). Also, and again because dt >> sr, the rear and front sight angle corrections are very nearly the same. The average of the two results is presented.

The sectional density of a bullet is computed according to the following formula using the bullet weight and diameter:

Sectional Density = bullet weight / [(diameter)(diameter)]

The English units of sectional density are lb/sq.in.

Muzzle Velocity calculations compensate for the effect of different barrel lengths. Enter the muzzle velocity at a known barrel length, that barrel length, your barrel length, whether or not rifle or pistol corrections apply, then click on calculate to see the actual approximate muzzle velocity for your firearm. the corrections are:

Actual Muzzle Velocity = Vref - (Lref - Lact)(K)

where,

Vref	=	known muzzle velocity
Lref	=	barrel length for known muzzle velocity
Lact	=	your actual barrel length
K	=	25 for rifles, 50 for pistols

Benchrest Shooting simulates repeat trajectories using a random variation of muzzle velocity, wind velocity and direction to determine expected group sizes. Enter the required four bullet parameters, diameter, weight, ballistic coefficient, and muzzle velocity. The program will make full use of the current Shooting Parameters as well. Enter also values for the standard deviation of muzzle velocity, wind velocity, and direction. If, for example, the muzzle velocity is 2600 ft./sec. and the standard deviation of muzzle velocity is 100 ft./sec. then the random muzzle velocity will fall within the range of 2500 to 2700 ft./sec. 68% of the time, and within 2400 to 2800 ft./sec. about 95% of the time. Enter also the number of shots to be included in the group.

The group size is the distance between the centers of the two shots the farthest apart at the target.

The target will be at the sight zero distance defined in the Shooting Parameters.

This calculation uses Greenhills's formula to determine the required barrel twist rate to produce a bullet rate of rotation that will properly stabilize it in flight. The formula uses the bullet's diameter and length as follows:

barrel twist rate = (150)(diameter)(diameter) / length

If the specific gravity of the bullet material is known, the twist rate is multiplied by a correction factor:

correction factor = Square Root(10.9 / Specific Gravity)

where 10.9 is the specific gravity of a jacketed lead bullet. Some other approximate specific gravity values are:

Lead	=	11.3
Copper	=	9.0
Aluminum	=	2.7
Tungsten	=	19.3

There are two ways to add a cartridge to the data file.

The 'Add New Cartridge' function brings up the quick data entry window. In this window you enter the cartridge type, bullet name, bullet weight, and data for each of the other data fields. Also, here the cursor will jump from field to field as you press the up and down arrow keys. Use the 'Add New Data' menu item in the File menu to add the data to the file. Lastly, you may have more than one 'Add Cartridge Data' window on the screen at a time.

The second way to add a cartridge to the database is directly in the analysis window. Modify an existing cartridge type and/or bullet name and/or bullet weight for the new cartridge in the data fields. At this point you can bring down the File menu and select 'Copy to New Cartridge ID' to store the new, modified cartridge in the data file as an additional entry. The data for the new cartridge can be added at any time by selecting each data field and modifying the existing numbers. Be sure to bring down the File menu again and select 'Save Changed Data' to store the data in the file if you do this after you made the copy.

To copy an existing cartridge, enter a new cartridge type and/or bullet name for the new cartridge in the data fields. At this point you can bring down the File menu and select 'Copy to New Cartridge ID' to store the new, modified cartridge in the data file as an additional entry. The data for the new cartridge can be added at any time by selecting each data field and modifying the existing numbers. Be sure to bring down the File menu again and select 'Save Changed Data' to store the data in the file if you do this after you made the copy.

There are two ways to delete a cartridge. You can select the cartridge from the main displayed list, then click on the 'Delete All Selected Cartridges' button at the bottom of the list. You can also delete a cartridge which is being displayed in its individual analysis window by bringing down the File menu and selecting 'Delete'.

To rename a cartridge, select a cartridge to rename from the main list, click the 'Work With Selected Cartridge Data' button to bring up the cartridge data and analysis window (or double click on its entry in the list). Enter a new cartridge type and/or bullet name and/or bullet weight, bring down the File menu, then click on 'Save As New Cartridge ID'.

Data items for each cartridge consists of seven fields.

After entering or modifying any data fields in this window, be sure to use the functions in the File Menu to save the changes to the database.

In addition to the above fields, sizing data for most cartridges is included in the data file. Click on the button marked 'Display Cartridge Data' in the 'Work With Cartridge Data' window to view this data.

The Display Cartridge Data Window depicts an actual size rendition of the cartridge along with dimension data. Two numbers point to various locations on the cartridge. The left number is the width of the cartridge at that point in inches. The right number is the height of the cartridge at that point.

Menu item options allow you to print the window or resize the window to show it actual size, two times actual size, or four times actual size.

You may press any of the buttons in the 'Work With Cartridge Data' window to compute a trajectory. Use the menu bar item entitled 'Bullet Plots x-Units' to specify if you prefer the x-coordinate axis for the plots to be either Distance or Time. For most of the plots, this also controls how the trajectory equations are integrated, using either the distance or time variable. This parameter also specifies the x-axis of the Bullet Spin Decay and Yaw of Repose Plots in the Custom Drag Function Window when using the Modified Point Mass Model. This does not control the x-axis for the 'Other Trajectory Analysis' plots. Use the menu bar item entitled 'Drop/Drift Units' for the amount of bullet drop or drift to be displayed in inches or feet for English Units or to be displayed in centimeters or meters for Metric units.

Bullet drop is the primary trajectory plot that you will observe to analyse the trajectories of bullets. Using the bullet data in the 'Work With Cartridge Data' window and the Shooting Parameters, the bullet is fired and progresses until the Analysis Distance or Analysis Time is reached. The plot produced depicts the bullet height on the y-axis and either distance or time on the x-axis.

Bullet drift plots indicate the amount of deviation from a straight line path that can be expected. For the Vacuum and Point Mass Trajectories, side drift will be due to Coriolis effects only. This will result in right hand drift in the Northern hemisphere and left hand drift in the Southern hemisphere. For the Modified Point Mass Model, bullet spin will cause an additional drift effect. Right hand (clockwise) bullet spin causes additional right hand drift. Left hand (counterclockwise) spin causes a drift to the left. A positive Barrel Twist rate in the Shooting Parameters window indicates a right hand twist, a negative value indicates a left hand twist.

These plots differ from the other plots in that either distance or time along the trajectory is shown on the y-axis and drift amount is shown on the x-axis. This provides a better depiction from the shooters point of view.

Bullet Velocity plots show the decay in bullet velocity along the trajectory. The x-axis units will be distance or time.

These plots show the bullet's kinetic energy (excluding spin energy) along the trajectory.

These plots show the bullet's momentum along the trajectory.

Other trajectory analyses are initiated by the other buttons. These include:

These are described in detail in separate sections.

The Aimpoint Correction plots show the point of aim deviation from the zeroed scope crosshairs or iron sights to hit a target at a range different from the zeroed distance. Assuming the sights are zeroed as defined in the Shooting Parameters, the Aimpoint Correction up to the Analysis Distance is shown in four plots.

The Aimpoint Correction plot always uses distance as the trajectory integration variable.

Maximum Point Blank Range shows two plots on the same graph. The x- axis is the usual target range. The y-axis is the lethal zone diameter. The lethal zone diameter means that the bullet will never be more than one-half the lethal zone diameter high and never less than one-half the lethal zone diameter low at any point along its trajectory to the target.

To use the plot, select a lethal zone diameter and move right until you come to the Sight Zero Line. Intersection of a selected lethal zone diameter with the Sight Zero Line indicates that the scope should be zeroed for that distance. The maximum Range Line indicates the bullet will be within the lethal zone at every point along to the trajectory up to that distance.

The Maximum Point Blank Range plot uses either distance or time as the trajectory integration variable.

Flight Time versus Distance shows how long it takes the bullet to fly along its trajectory. Downrange distance is shown on the x-axis, flight time is shown on the y-axis.

The Flight Time versus Distance plot uses either distance or time as the trajectory integration variable.

The Maximum Range plot shows the maximum range and height reached by the bullet for various initial launch angles. The x-axis shows the muzzle angle from 0 to 90 degrees. The y-axis shows the range or height reached. For most projectiles in the atmosphere, the maximum range is obtained at a launch angle of about 32 degrees. In a vacuum, the best angle would be 45 degrees.

The Maximum Range plot always uses time as the trajectory integration variable.

The Distance Trajectory Table produces a tabular presentation of the trajectory data. Shown at 10 yard (or 10 meter) intervals is the trajectory data, flight time, bullet drop, drift, velocity, energy, and momentum. For the Modified Point Mass model, a bullet spin rate column is added.

Use the File menu functions to print the table or export the data to a file.

The Distance Trajectory Table always uses distance as the trajectory integration variable.

The Time Trajectory Table produces a tabular presentation of the trajectory data. Shown at .01 second intervals is the trajectory data, downrange distance, bullet drop, drift, velocity, energy, and momentum. For the Modified Point Mass model, a bullet spin rate column is added.

Use the File menu functions to print the table or export the data to a file.

The Time Trajectory Table always uses time as the trajectory integration variable.

The Shooter's View Aimpoint Function presents a graphical scope image of the target and required aimpoint. The scope is assumed to be zeroed at the distance specified in the Shooting Parameters. The target is assumed to be at the Analysis Distance also specified in the Shooting Parameters. The image shows the required hold over aimpoint you should use to hit the displayed target. With a specified crosswind, the crosshairs will show the windage as well as the required elevation correction. Other lateral corrections are due to Coriolis and spin drift effects. The image shows a target stand 5 feet tall and, initially, a 12 inch wide target. Each of the colored bands on the target have a width dependent on the target size. For a 12 inch target, they are each two inches wide. For a 48 inch target, they are 8 inches wide.

The initial magnification factor is 10X, you may use the menu items to adjust this value in various values down to 1X and up to 30X. The image is initially displayed through a duplex reticle. You may choose to bring up a Mil-Dot, Moa-Dot, or Bullet Drop Compensating reticle.

The Mil-Dot reticle is a Leupold style with the thick lines being 10 milliradians apart. The advantage of the computer reticle is that the Mil-Dots are correct at all magnifications unlike with real scopes where the Mil-Dots are calibrated only at the highest magnification. The center of the ovals on the reticle are 1 milliradian apart. One milliradian subtends 3.6 inches at 100 yards.

tangent(.001 radians) x 100 yards x 36 in. per yard = 3.6 in.

The ovals are .25 milliradians wide and the lines between the ovals are .75 milliradians long. The oval heights are arbitrary being of no particular height. A Mil-Dot reticle is useful for determining the range to a target of known height or width. The height of the target in yards divided by the height of the target in milliradians times 1000 gives you the range to the target in yards. A three-foot tall target (1 yard) which subtends four dots on the reticle, is 1/4 times 1000 = 250 yards away.

The Moa-Dot reticle is similar to the Mil-Dot Reticle except the center of the dots are one minute of angle (MOA) apart and there are more of them. One minute of angle is 1/60 of a degree and represents approximately (but not exactly) 1 inch at 100 yards. The thick lines on the reticle here are 20 minutes of angle apart.

The Bullet Drop Compensating reticle shows several crosshairs below the center crosshair. They provide aimpoints for targets at various additional ranges assuming the scope is zeroed for 100 yards or meters (depending if you have selected English or Metric units). The center crosshair assumes a zero of 100 yards or meters and the additional crosshairs should then be used for targets at the other distances shown on the reticle. Depending on the magnification, which limits the field of view, distances up to 1000 yards or meters will be shown in 100 yard or meter increments.

You may adjust the target size in selected values ranging from 6 inches to 48 inches.

The Shooting Parameters window provides access to a number of values that control the trajectory. These are described in a separate help window.

The Drag Function menu contains features to produce a plot of the currently selected drag function (see Field 17 in the Shooting Parameters description help window) and a plot of all drag functions on a single graph for comparison. This will plot all classic stored drag functions plus the currently selected custom drag function if that is the one selected.

You may also show or hide the Custom Drag Function Window (see that section).

The File menu includes a function to reset the shooting parameters to their default values. This also stores the changes automaticaly to the database.

Be sure to use the Save Data function in the File menu to save any changes you make to the Shooting Parameters.

The Shooting Parameters window contains numerous data fields that define various aspects of the trajectory calculations. There are 23 fields here.

JBallistics is supplied with drag function data for the G1, G2, G5, G6, G7, G8, GS, and Ingalls shape projectiles. The shape of these projectiles is stored for your view using the Custom Drag Function Graphics Tools. GS is the drag function for a 9/16" sphere. The Ingalls projectile is the same shape as the G1 but has slightly different drag data due to different techniques used at the time of measurement.

JBallistics is supplied with custom drag functions for:

This window is made visible and hidden from a menu item available in the Shooting Parameters Window. The Custom Drag Function window cannot be closed.

Menu items here control the use of the aerodynamic coefficients for the Modified Point Mass algorithm. You may use the built-in models for predicting them or use tabulated data supplied and entered by you. The table for entering and maintaining these coefficients is also made visible and hidden using the menu item here. You may have a separate table of coefficients for each custom drag function maintained in the database.

You may also produce various plots of the drag function, aerodynamic coefficients, bullet spin decay, and quasi steady-state yaw of repose for the most recent analysis using the Modified Point Mass Model.

This window also contains the functions for maintaining the database of custom drag functions. These functions are described in detail in other sections.

Here you specify the bullet dimensions for use by the built-in aerodynamic coefficient models including the computation of custom drag functions. You may specify custom drag functions for use by the Point Mass Model as well. The button labeled 'Compute and Plot Drag Function' takes the bullet dimensions above it and computes the custom drag function for you to view. It is not necessary to click on this button for the program to use the drag function. This is done automatically when the drag function is selected in the Shooting Parameters window.

At the bottom of the window is a data field for 'Your Screen Height'. This is used by the Bullet Shape Graphics Tool to depict the bullet actual size on your computer monitor. This parameter is saved in the data base using the 'Save Data' menu item in the Shooting Parameters Window. It is displayed here because this is a more logical place to display it. The button at the bottom of the window initiates the Bullet Shape Graphics Tool for the bullet dimensions displayed.

To add a custom drag function, simply modify the name of the currenly displayed drag function, modify the data fields as you desire then bring up the menu items under the 'File' menu item and select either 'Add New Function' or 'Copy to New Function Name'. A new entry will appear in the drag function list.

To copy a custom drag function, simply modify the name of the currenly displayed drag function, modify the data fields as you desire then bring up the menu items under the 'File' menu item and select 'Copy to New Function Name'. A new entry will appear in the drag function list. Remember, when you copy a custom drag function you are also copying any tabular aerodynamic coefficients associated with it.

To delete a custom drag function, bring up the function for display by double clicking on its name or single clicking on its name followed by a click of the 'Work With Selected Function Data' button. Next , either bring up the menu items under the 'File' menu item and select 'Delete' or click on the 'Delete Selected Function' button. You will be asked to confirm the delete.

To rename a custom drag function, simply modify the name of the currenly displayed drag function, modify the data fields as you desire then bring up the menu items under the 'File' menu item and select 'Save As New Function Name'. The entry will appear in the drag function list with the new name. Do not select (with a single mouse click) a different entry in the drag function list in between the time you brought up a drag function for display and then changed its name field. If you do this the selected drag function will be renamed, not the one you brought up orginally.

There are eleven data fields that define the bullet shape parameters for purposes of determining its aerodynamic coefficients. They are:

The Bullet Shape Graphics Tool lets you draw an actual size bullet on the screen. The bullet shape parameters form the basis for predicting the aerodynamic coefficient using the Modified Point Mass Model. You may also formulate a custom Drag Function only for use by the Point Mass Model. The initial drawn shape is determined by the 'Bullet Data for Drag Function' data in the Drag Function Window. The 'Your Screen Height' value allows JBallistics to draw the bullet at its actual size. Bring up the Bullet Shape Graphics Tool by clicking on the 'Display Bullet Shape Graphics' button at the bottom of the Drag Function Window.

The bullet shape is determined by the position of nine sliders displayed below the bullet graphic in the window. As you move these sliders by dragging the mouse while holding down the left mouse button, the actual numerical value of the parameter is updated both in the window next to the slider and in the Drag Function window. The bullet shape is altered on the screen as you move the slider. You may effect more precise movements of the slider with single mouse clicks on each side of the slider control indicator.

While you may move any slider to any position, the physical geometry of a bullet has to be self consistent. That is, it may not be possible to draw a bullet with all these parameters as they are currently being defined. A message on the bullet graphic portion of the window will indicate that JBallistics is unable to draw the bullet body or bullet nose. Often, a particular parameter will permit a bullet to be drawn only within a very narrow range of permissible values. In short, you have to experiment and move the sliders around carefully and precisely to obtain the bullet shape you need. You may, in fact, want to hold up your bullet next to the screen until you obtain the desired match.

See the section entitled 'Custom Drag Function Bullet Data' for a description of each of the bullet shape parameters.

If you move the Meplat Diameter slider to a positive value, the Nose Radius and Ogive Center Offset sliders will be automatically reset to zero and, while they may subsequently be moved, will have no effect. This is because those two parameters are redundant if the bullet tip has a positive Meplat area.

When you are finished with the drawing, you may use the 'File' menu to store the changes to the database.

The JBallistics Modified Point Mass model requires seven aerodynamic coefficients to be specified. These are listed in approximate order of significance:

All of these are actually functions which vary by mach number.

JBallistics contains advanced prediction algorithms to estimate these based on the dimensions of the bullet, its center of gravity, and its axial and transverse moments of inertia. JBallistics uses the dimensions of the bullet to estimate these latter three values.

The Aerodynamic Coefficients Table allows the user to enter by hand known values of the aerodynamic coefficients for certain mach numbers. The table is brought up by specifying 'Show Coefficient Table Window' in the Custom Drag Function Window. The table is removed by specifying 'Hide Coefficient Table Window' in the drag Function Window. While there can only be one table, its contents depend on the particular custom drag function selected. Each custom drag function can have a set of tabular coefficients associated with it.

The menu items for the table contain functions to store the data in the data base following modifications or data entry. You may also clear all the data fields or print the table. Another function allows you to import into the table the values computed by JBallistics during the most recent analysis in which the program was called upon to compute them. This was when 'Use Calculated Coefficients' was selected in the drag function window and a trajectory was computed.

After a trajectory analysis has been performed, the 'Last Analysis' menu will be activated in the Custom Drag Function Window. For the POint Mass Trajectory, the first menu item entitled 'Drag Coefficient, Cd0' will be available. Here you may produce a plot of the drag function used in that most recent analysis. Cd0 is the zero yaw drag coefficient.

For Modified Point Mass trajectories, a host of other plots are available. You may produce a plot of the computed (or from a table if so specified) aerodynamic coefficients:

Four other plots will be available:

The zero yaw Drag Coefficient is a measure of atmospheric retardation of the forward motion of the projectile. This plot shows the retardation coefficient for a projectile of a certain shape, but not its size. Total drag should incorporate the drag component due to the yaw angle of the projectile. The yaw coefficient is shown in a separate plot. The total drag will be:

Cd = Cd0 + (Cd2)(sin(a)sin(a))

where,

Cd0	=	zero yaw drag coefficient
Cd2	=	yaw drag coefficient
a	=	total yaw angle

This coefficient has a significant effect on the bullet trajectory.

The Yaw Drag Coefficient is a measure of atmospheric retardation of the forward motion of the projectile due to that component of the projectile face shown to the air that is offset from a nonzero yaw angle.

total drag will be:

Cd = Cd0 + (Cd2)(sin(a)sin(a))

where,

Cd0	=	zero yaw drag coefficient
Cd2	=	yaw drag coefficient
a	=	total yaw angle

This coefficient has only a minor effect on the bullet trajectory.

The Lift Coefficient is a measure of the force that is perpendicular to the trajectory and tends to pull the projectile in the direction its nose is pointed. This coefficient has a significant effect on the bullet trajectory.

The Spin Damping Moment Coefficient is a measure of the forces acting on the bullet that tend to dampen its spin rate over time. This quantity is always negative. This coefficient has a significant effect on the bullet trajectory.

The Pitching, or Overturning Moment coefficient, Cma0, describes the tendency of the yaw angle of a trajectory to increase for a positive value or decrease for a negative value. The total coefficient also incorporates the Cubic Pitching (Overturning) Moment Coefficient. This coefficient has a significant effect on the bullet trajectory.

The Cubic Pitching, or Overturning Moment coefficient, Cma2, describes the nonlinear behavior of the total Pitching (Overturning) Moment Coefficient. The total coefficient incorporates both the linear term and the cubic coefficient as follows:

Cma = Cma0 + (Cma2)(sin(a)sin(a))

where,

Cma0	=	linear pitching (overturning) moment coefficient
Cma2	=	cubic pitching (overturning) moment coefficient
a	=	total yaw angle

The coefficient is referred to as the cubic coefficient even though it contributes to the total coefficient with the square of the yaw angle because the Lift or Normal Force contains an additional sin(a) term multiplied by the coefficient which produces a cubic dependency by the cubic coefficient. This coefficient has only a minor effect on the bullet trajectory.

The Magnus Force Coefficient describes the force on the projectile due to unequal pressures on each side of the projectile caused by its rate of rotation. This coefficient has only a minor effect on the bullet trajectory.

The Bullet Spin decay plot shows the rate of rotation of the projectile as a function of downrange distance or flight time.

The Yaw of Repose is a quasi-steady state yaw angle predicted by the Modified Point Mass Model. This yaw angle forms the basis of the spin drift computations.

The stability factor shows the value of the factor as a function of mach number. A factor is 1.5 is generally required to ensure stability of the bullet at all points in its trajectory. If the plot dips below 1.5 anywhere then the stability of the bullet is not assured at that point and it may begin to tumble.

The Required Twist Rate, shown as a function of mach number, indicates the barrel twist rate required to launch a trajectory that will be stable at all points in its flight. Or it can indicate where instability is likely to occur if your barrel twist rate is less than a value shown on the plot.