Range Report Looking for an equation

[carswellb58]

I am interested in plotting the exact (theoretical) path of a bullet and could use some help.

I am confident that I can generate a plot of the projectile's height as a function of time, but I am having a bit of difficulty determining the equation to plot the projectile's distance down range as a function of time. That is, if the x-axis is time and the y axis is distance traveled, what equation correctly describes that relationship.

Once I have that equation I can use graphing software for plotting parametric equations to generate the desired final graph, that being one where the x-axis is distance traveled and the y-axis is height.

I'd welcome any insight that could be offered. I do have a rough equation, but I fear it is a bit crude. I'm looking for the best refinement I can find.

Thanks,

John

 

[Macht]

If you want the "best refinement" you can find, you'll be looking at a full 6-DOF solver, or you could compromise slightly and use the MPMM. Modern Exterior Ballistics by McCoy or Ballistics: Theory of Design of Guns and Ammunition by Carlucci and Jacobson will give you everything you want. Accurate calculation of trajectories is non-trivial.

Alternatively, you should be able to access the original papers the books reference. 6-DOF MPM

 

[wadebrown]

The previous poster gave you some great advice, directly to the sources. If you have had some numerical analysis, especially solving ordinary differential equations the second reference listed above will be enough aid to achieve what you want. In the event that the last sentence sounded like a lot of greek use JBM Ballistics solver.

 

[Nik H]

I am interested in plotting the exact (theoretical) path of a bullet and could use some help.

I am confident that I can generate a plot of the projectile's height as a function of time, but I am having a bit of difficulty determining the equation to plot the projectile's distance down range as a function of time. That is, if the x-axis is time and the y axis is distance traveled, what equation correctly describes that relationship.

Once I have that equation I can use graphing software for plotting parametric equations to generate the desired final graph, that being one where the x-axis is distance traveled and the y-axis is height.

I'd welcome any insight that could be offered. I do have a rough equation, but I fear it is a bit crude. I'm looking for the best refinement I can find.

Thanks,

John

You could do all of the above or you could simply look in the resource section at Jim's article

Trajectory of a Rifle Bullet

He is a member of very high regard in this area

 

[bluegrasspicker]

OP, you said you're "looking for an equation."

The drag force on the bullet changes with the bullet's speed in a way that isn't simple. So "simple" physics with simple math (where we get to use an equation) isn't going to be fruitful in describing the trajectory .

As the eggheads are apt to say, "there is no closed-form analytical solution."

That's why the folks above are suggesting numerical integration software. Roughly speaking the process is to numerically calculate the force on the bullet, the resulting acceleration, then the resulting velocity, then the resulting change in position. The computer uses a small time step and repeats the process a bazillion times since the force on the bullet is changing each time. So I guess it really is somewhat simple physics and math, it's just not in the form of an equation.

 

[Kwait33]

As the others have stated these are fairly complex equations.

However, since you state that you want to plot the the path in 3D, why not use ballistic solvers that are already out there and use steps small steps for the output? This will generate plenty of data, in which you could plot the drop and horizontal movement as function of tof (Time of Flight).

Here is a plot of a 30 cal 168 BTHP using Hornady 4DOF plotted with drop (X) and horizontal movement (Y) ( spin drift, no wind) over time (Z) (distance of 2500 yards), drop is calculated in inches for every yard. ( Really rough plot from R)