- #1

Zaggy

- 1

- 0

**Aim:**

Having established a set of initial conditions, I would like to be plot the trajectory of a ballistic object (in two dimensions, vertical and horizontal) at a repeating interval (say every 1/100th or 1/1000th of a second), taking into account gravity (obviously), accurate drag model & ballistic coefficent and the effects of an atmosphere (initially, the standard atmosphere)

**Initial Conditions:**

*i) the 'world' / standard atmosphere*

We can assume a standard world where g will be constant, though RHO (and speed of sound) may have a different value OR be changing during flight (ie as a projectile is shot upwards into thinner/colder air):

*g*(gravity) : -32.15 ft/s^2

*h*(altitude) : 0 ft

*T*(Temperature) : 59deg F (where

*T*= 59 - 0.00356 *

*h*)

*p*(pressure) : 2118.145 lbs/sqft (where

*p*= 2116 * (((

*T*+ 459.7) / 518.6) ^ 5.256))

*RHO*(density) : 0.0023769 slugs/cuft (where

*RHO*=

*p*/ (1718 * (

*T*+ 459.7)))

Speed of Sound : 1119.28 ft/s (assuming 80% humidity)

*ii) projectile properties*

We are assuming a spin-stabilised projectile, with a circular cross-section (I'm trying to use a real-world set of numbers, based up high-power target shooting projectiles) that is fired horizontally:

*v*(velocity) : 3000 ft/s

*W*(weight) : 180grains

*r*(radius of cross-section) 0.154 inches (ie, a .308" cal projectile)

*BC*(Ballistic Coefficent) : 0.400 lbs/in^2 (G7 Drag Model - see excerpt)

'G7' Drag Model excerpt (30 data points, instead of 80 - Drag Model assumes the Standard Atmosphere is present; so as I understand it, we don't even really need

*h*,

*T*,

*p*and

*RHO*; just a means to convert the current

*v*(ft/s) into

*Mach*; which is not difficult):

**Mach CDrag**

0.900 0.1464

0.925 0.1660

0.950 0.2054

0.975 0.2993

1.000 0.3803

1.025 0.4015

1.050 0.4043

1.075 0.4034

1.100 0.4014

1.150 0.3955

1.200 0.3884

1.300 0.3732

1.400 0.3580

1.500 0.3440

1.600 0.3315

1.700 0.3209

1.800 0.3117

1.900 0.3042

2.000 0.2980

2.100 0.2922

2.200 0.2864

2.300 0.2807

2.400 0.2752

2.500 0.2697

2.600 0.2643

2.700 0.2588

2.800 0.2533

2.900 0.2479

3.000 0.2424

**Method:**

Now, as above, the goal is to plot the movement of a project over time in 2D space. Plotting a trajectory with ONLY gravity is easy - plotting a trajectory with a CONSTANT retardation is easy (say 60% retardation per second, which is 'in the ballpark' for this sort of projectile) - BUT I have no idea about how to go about figuring out the retardation using the Ballistic Coefficent and the Drag model (obviously the gravity component is not a problem, just 'deceleration' or 'retardation' of the projectile)...

In otherwords, what do I need to in order to calculate velocity change from time= 0.000 to time = 0.001... eg:

*t*= 0.000,

*v*= 3000.0 -->

*t*= 0.001,

*v*= 3000.0 -

*x1*-->

*t*= 0.002,

*v = (3000.0 -*

I hope that makes sense to someone out there :)

Cheers

Zaggy

*x1*) -*x2*; etc...I hope that makes sense to someone out there :)

Cheers

Zaggy