6 2 KINEMATICS. [122. 



of the acceleration at any distance OP = s(<R), we have 

 j\g=s \R t or j=gs/R. But the acceleration tends to diminish 



the distance s, hence -= ^s. Denoting the positive cori- 

 dt R 



stant g/R by ft 2 , the equation of motion is 

 ;'' g=-A where ^ = ^L (19) 



Integrating as in Arts. 117 and 119, we find 



If the particle starts from rest at the surface, we have v=& 

 when s = R ; hence o = J p?R z + C ; and subtracting this from 

 the preceding equation, we find 



S*, (20) 



where the minus sign of the square root is selected because 

 s and v have opposite sense. 



Writing ds/dt for v and separating the variables, we have 



whence /=-cos~ 1 ^ s + C r . 



As s=R when /=o, we have o=-cos~ 1 i + C r , or 



f* 

 Solving for s, we find 



(21) 

 Differentiating, we obtain v in terms of t\ 



(22) 



