Development Of A Mathematical Ballistics Model Using Differential Equations (Part 2)

 We left off in Part 1 with a conversion problem to solve before proceeding. That was to change the relevant transformation equations (6), e.g.

x =   r cos q

y =  r sin q

r²  =  x²   +  y ²

To polar coordinate form.  Repeatingn the instructions:

The procedure entails differentiating the transformation set (6) twice with respect to t and then substituting the results into equations (5).

So we first need to obtain:

   d²x/ dt ²    = d²/dt ²  (r cos q)  =  x''

   d² y/ dt ²    =    d²/dt ²  (r sin q) =  y''

Then we obtain:

x''   =   r'' cos q   -   2r' q' sin q   - r q'' sin q  -  r q'² cos q   

y''   =   r'' sin q   -   2r' q' cos q   - r q'' cos q  -  r q'² sin q   

Which results now must be substitutes into equations (5):

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

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

From which we obtain:

(9a) A _x = x'' =  

 r'' cos q   -   2r' q' sin q   - r q'' sin q  -  r q'² cos q   = - cr^-2   cos q

(9b) A _y = y'' = 

r'' sin q   -   2r' q' cos q   - r q'' cos q  -  r q'² sin q =  - cr^-2   sin q

These equations can now be simplified by multiplying (9a) by cos q  and (9b) by sin q then adding:  Doing so we get:

r''  -   r q'² =    - cr^-2 

Now, multiply the first equation (9a) by sin q  and the second equation (9b) by   cos q and subtract the first from the 2nd to get:

4 r' q'    + 2 r q''     =  0

Then the two equations:

(10)

r''  -   r q'² =    - cr^-2 

And:

4 r' q'    + 2 r q''     =  0

Describe the motion of the particle m in polar coordinates.  The acceleration being along the line joining M and m and normal to this line, i.e.

The equation sets (7) or (10) then form the mathematical model for the preliminary ballistics problem.  But equations (10) as we'll see in Part 3 is more apropos to solving the full ballistics model.