2''

Let  AD  be  the  vertical  diameter  of  a  circle  and
AB  =  I  any  chord.  Join  DB.  Then  we  have  AB
=  AD  cos  DAB  =  AD  sin  ABC.     If  AB  =  I,  we

B    have  also  AB  =  ^ff  sin  ABC.     Therefore  AD  =

\ft^,  or  t  =|/^^.  This  is  independent  of  the  posi-
tion of  the  chord  AB,  and  therefore  t  is  the  same  for  any  chord

through  A  or  D.
•  '  151


152  KINEMATICS   OF   A    POINT — TRANSLATION.         [CHAP.  V,

Hence  for  uniform  acceleration  /,  the  time  of  descent  down  alt
chords  through  the  highest  and  lowest  points  of  a  circle  are  equal.
This  property  enables  us  to  find  the  line  of  swiftest  descent  to  a
given  curve  from  any  point  in  the  same
vertical  plane.
Thus  if  EG  is  the  curve  and  A  the  pointy
draw  AO  vertical  or  parallel  to  the  direc-
tion of/,  and  with  centre  in  AC  describe  a
circle  passing  through  A  and  tangent  to
the  curve  EG  at  P.
Then  AP  is  the  line  of  swiftest  descent
from  A  to  the  curve  EG.  For  any  other
point  p  in  EG,  Ap  cuts  the  circle  in  some  point  q,  and  since  the
time  from  A  to  g  is  equal  to  that  from  A  to  P,  the  time  from  A  to-
p  is  greater.
EXAMPLES.
g  =  32.16  ft.-per-sec.  per  sec.     Friction,  etc.,  disregarded.
(1)  Find  the  position  of  a  point  on  the  circumference  of  a  vertical
circle,  in  order  that  the  tim£  of  rectilinear  descent  from  it  to  the
centre  may  be  the  same  as  the  time  of  descent  to  the  lowest  point.
Acceleration  due  to  gravity.
Ans.  30°  from  the  top.
