Bullet deceleration due to drag

[Raddy13]

My friend and I are working on a realistic FPS sniper simulator and we wanted to be able to realistically model things like bullet drop, flight time, and impact energy. I've managed to find the drag coefficients for different rounds and I know the exit velocity, so now I want to use that to model the bullet's trajectory. Here is a sample using a .22 bullet:

$$C_d = 0.169\\ v_i = 330~m/s\\ m = 0.00259~kg\\ A = 2.45e\!- 5~m^2\\ \rho_{air} = 1.204~kg/m^3\\$$

$$F = m\cdot a\\ F_d = \frac{1}{2} \rho v^2 C_d A\\ v_f = v_i + a\cdot t$$

I can set this up to be solved in discrete time-steps, but I can't figure out how to get a continuous solution since drag is dependent on the velocity and the velocity depends on the drag. I found a few solutions for determining terminal velocity for an object in free-fall, but I can't figure out how to apply that here; it's been a long time since Diff. Eq. Any thoughts?

 

[Jupiter6]

Try googling "projectile motion with air resistance".

 

[rcgldr]

You can start with this wiki article:

http://en.wikipedia.org/wiki/External_ballistics

 

[Raddy13]

Thanks for the replies, guys. Apparently the general solution is obnoxiously complex so most simulations just do it with discrete time steps. For anyone looking for information on how to do that, this link gives a pretty good explanation.

Thanks again!

 

[PJC01]

I would suggest using numerical methods such as Euler's method or Runge-Kutta methods to solve for the bullet's trajectory. These methods use discrete time-steps to approximate the continuous solution and can handle situations where the equations are dependent on each other, such as in the case of drag and velocity. Additionally, these methods can be easily implemented in computer simulations, making them ideal for your FPS sniper simulator.

Another approach could be to use analytical solutions, such as the drag equation or the ballistic trajectory equation, to model the bullet's trajectory. These equations take into account factors such as drag, initial velocity, and mass, and can provide a continuous solution for the bullet's motion. However, they may not be as accurate as numerical methods, especially for complex scenarios.

Furthermore, it may be helpful to consult with a mathematician or physicist with expertise in differential equations to assist with implementing a continuous solution for the bullet's trajectory. They may be able to provide valuable insights and guidance on how to approach the problem.

Overall, accurately modeling bullet drop, flight time, and impact energy in your simulator will require a combination of numerical and analytical methods, as well as collaboration with experts in the field. I wish you the best of luck in your simulation project.