84 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



134. A particle moves in a nearly circular orbit with an acceleration 

 ji + i(r a), a being the mean radius; show that the apsidal angle is 

 ir<o/J(3a> 2 + v), where a> is the mean angular velocity. 



135. If the central acceleration is p.u^ the velocities at the two apsidal 

 distances satisfy the relation z^ 2 + # 2 2 = 2A 4 //M. 



136. A particle describes a central orbit with acceleration p (r~ 5 -^a 2 r~ 7 } 

 starting from a point where r=a with velocity |- <JC2fjC)/a 2 at an inclination 

 sin" 1 4 to the radius vector. Prove that its path is 



137. A particle describes a central orbit with acceleration /i/(r-a) 2 

 towards the origin, starting with the velocity from infinity from a point at any 

 distance c between a and 2a at an angle 2 cos" 1 \/(a/c). Prove that the path 

 described is given by the equation 



r-a -tan- 1 r-aa. 



138. A particle moving with a central acceleration 4* 2 (2a~ 3 - 3ra~ 4 2r 3 a~ c ) 

 starts from a point distant j^a from the origin in direction making an angle 

 tan -1 27/125 with the radius vector with such velocity that the rate of descrip- 

 tion of areas is K. Show that the equation of the orbit is 



