APPENDIX A

Transformation of the Equation of Free Planar Motion to the Axes of D0

As a further example of the relationship between D1 and D0, the equation of planar motion in D1 (3.18) is, in this appendix, transformed to the axes of Do.


Transformation of the term 
××
s
 
 in (3.18) is given in terms of u by (4.24). Transformation of the other terms is as follows.



From (2.19) and (4.18)

u2w2s = æ
è
df

dt
ö
ø
2

 
( r + a )
(A.1)

and from (2.19), (4.12) and (4.18)


×
s

2

 

u


du

ds
 =  2

u3
æ
è
dr

dt
ö
ø
2

 
du

dr
(A.2)

and finally from (4.12)

- c2u du

ds
= - c2u du

dr
(A.3)

The full transformation of (3.18) in terms of u then becomes, from (4.24), (A.1), (A.2) and (A.3)

d2r

dt2
= - c2u3 du

dr
+ 3

u
æ
è
dr

dt
ö
ø
2

 
du

dr
+ u2 æ
è
df

dt
ö
ø
2

 
( r + a )
(A.4)

Substitution of (4.7) and (4.18) for u and s, then expands this to the final result

d2r

dt2
= - a c2 ( r - a )

( r + a )3
+ 3a

r2 - a2
æ
è
dr

dt
ö
ø
2

 
+ ( r - a ) æ
è
df

dt
ö
ø
2

 
(A.5)


G1 Version 2.2.4
Ó P.G.Bass, November 2009
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