MOT 



Ex. 4. If a body has fallen t", the space described in the last n" =z 



2 

 Ex. 5. Space described in the second immediately previous to the last 



Force variable. 



~ Tf "" "" J -~d~T 



When the force is a function of the space, we may hence find the velo- 

 city and time. For multiplying the above equations crossways, to elimi- 

 nate dt, 



vdv = fds. 



If we put for /its value in terms of s, we can integrate the last equa- 

 tion, and thus obtain | 2 fi.fds, whence v is known. 

 After this t is determined by the equation 



Ex. 1. Let a body fall from rest from the distance a towards a centre 

 of force varying directly as the distance ; to determine the motion. 

 Let m be the absolute force tending to the centre, then 

 f= mx t .". vdv = mxds =. mxdx .", 02 = C mx$ 



m (a2 x*}. 



And v wi (as ,r8)a 



/. t = . arc (cos. = j 



Ex. 2. Required the same when /varies as -^-. 



f '2m fa x 



.-. - V __ x v . 



v ' ' v 2m * 7( - 

 (arc -f sin.) whose ver. sin, is a x, 



187 



