Ballistics differential equations

To solve for v, you would take the reciprocal of both sides: v= \frac{-2M}{DBA}t+\frac{1}{C} In summary, the conversation discusses using a specific equation to calculate the acceleration and speed of a bullet based on various factors such as air density, velocity, ballistic coefficient, cross-sectional area, and mass. The equation can be manipulated to graph acceleration vs time and then integrated again to determine velocity and displacement vs time. The conversation also mentions that this process will be further explored in a math class and the main focus is on obtaining the answer for a program. A summarized equation is also provided.
  • #1
pousser
1
0
i have wandered past my math background into drag equations on bullets. i found the equation that relates acceleration and speed of the bullet. I do not know how to integrate multi-varible equations. How to i get the equation so it can be graphed on an acceleration vs time? I am thinking from there i can integrate again to velocity vs time then again to displacement vs time. I am not really interested in learning how to do this right now, since i will learn it in class later, i just need the answer to complete my program. The equation i have is
a=(D*V^2*B*A)/(2*M)

D=air density
V=velocity
B=ballistic coefficient
A=cross sectional area
M=mass
Thanks for your time
 
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  • #2
So you have
[tex]\frac{dv}{dt}= \frac{DBA}{2M}v^2[/tex]

That separates as
[tex]\frac{dv}{v^2}= \frac{DBA}{2M}dt[/tex]

which can be integrated to
[tex]-\frac{1}{v}= \frac{DBA}{2M}t+ C[/tex]
where C is a constant of integration.
 

1. What are ballistics differential equations?

Ballistics differential equations are mathematical models that describe the motion of projectiles, such as bullets or missiles, under the influence of gravity and air resistance. These equations take into account factors such as initial velocity, mass of the projectile, and angle of launch.

2. How are ballistics differential equations used in real life?

Ballistics differential equations are used in fields such as military technology, sports science, and forensic science. They are used to predict the trajectory and impact of projectiles, design weapons and ammunition, and analyze crime scenes involving firearms.

3. What are the key components of a ballistics differential equation?

The key components of a ballistics differential equation are the initial conditions, such as the initial velocity and angle of launch, and the forces acting on the projectile, such as gravity and air resistance. These components are used to solve for the position, velocity, and acceleration of the projectile at any given time.

4. How are ballistics differential equations solved?

Ballistics differential equations are typically solved using numerical methods, such as the Euler method or the Runge-Kutta method. These methods involve breaking down the equations into smaller steps and using approximations to calculate the position and velocity of the projectile at each step.

5. What are the limitations of ballistics differential equations?

Ballistics differential equations do not take into account factors such as wind, temperature, and air pressure, which can affect the trajectory of a projectile. Additionally, they assume a uniform and symmetrical shape for the projectile, which may not always be the case in real-life scenarios. Therefore, these equations may not always provide an accurate prediction of the actual flight path of a projectile.

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