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



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

 (ra), a being the mean radius; show that the apsidal angle is 



+ j&amp;gt;), where a&amp;gt; is the mean angular velocity. 



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

 distances satisfy the relation v^ + v% 2 = 2A 4 //*. 



136. A particle describes a central orbit with acceleration /* (r~ b --|a 2 r~ 7 ) 

 starting from a point where ra with velocity f v/(2jLi)/a 2 at an inclination 

 sin&quot; 1 4 to the radius vector. Prove that its path is 



137. A particle describes a central orbit with acceleration /u/(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&quot; 1 \/(a/c). Prove that the path 

 described is given by the equation 



\B = tanh&quot; 1 V {(r - a) /a} - tan - 1 J{(r - )/} . 



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

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

 tan&quot; 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 



