Development Of A Mathematical Ballistics Model Using Differential Equations (Part 1)

 

                                                              Fig. 1. Ballistic trajectories

                                                 Fig. 2.  Preliminary Mathematical Model

Ballistic trajectories have wide applications in our modern world, from lobbing missiles at enemy states to launching near Earth space craft. To proceed with the development of the mathematical model  (Fig. 1)we first  need to develop the basic equations of motion, stating with the much simpler Fig. 2.   Here M is a mass fixed at the origin in the x-y plane with a particle of mass m situated at the point P in the plane.  The rectangular coordinates of P as shown are (x,y).  The polar coordinates are (r, q).  According to Newtonian mechanics the mutual attraction between two masses, M and m, is given by the inverse square law:

1) F _Mm = GM m/ r²

Where F is the force vector and r is the distance vector.   If we assume the mass m is negligible compared to M, the force exerted on m is given by: 

2) F _m = GM A

Where A  is the acceleration imparted to m relative to the x-y coordinate system.  Since equations (1) and (2) are equivalent expressions of the same force, we can write:

3) F _Mm = F _m  or:   cm/ r²   =  m A

And the vector A  is anti-parallel to r.   Then the scalar of  A is given by:

4) A =  - c/ r²   

The components of acceleration from Fig. 2 are:

5)

A _x = x'' =  A cos q   =  - c/ r²   cos q

A _y = y'' =  A sin q   =  - c/ r²   sin q

Note equations (5) display a mix of rectangular and polar coordinates. Obtaining correct equations in rectangular and polar coordinates alone requires use of the following elementary transformations:

(6)

x =   r   cos q

y =  r sin q

r²  =  x²   +  y ²

To then obtain the equations in rectangular coordinates alone we solve for  sin q , cos q  and r then substitute into equations (5) to get; 

(7)

x''  =   - cx (  x²   +  y ²  ) ^3/2

y'' =  - cy (  x²   +  y ²  )  ^3/2

To then obtain the equations in polar coordinates the procedure entails differentiating the transformation set (6) twice with respect to t and then substituting the results into equations (5).

Problem to lead off next part:

Perform the procedures as described above to obtain the polar coordinates form of the equations.