EXAMPLES. 77 



73. A particle P describes a rectangular hyperbola with an acceleration 

 pCP from the centre C ; a point Y is taken in CP so that CP. CY=a? ; prove 

 that the rate at which P and Y separate is 



where 2a is the transverse axis. 



74. If the acceleration of a particle is directed to a point and varies N/ 

 inversely as the square of the distance, prove that there are two directions 



in which it can be projected from a point P so as to pass through a point Q> 

 and that the velocity of arrival at Q is the same for both. Prove also that 

 the angle between one of the directions of projection and PQ is the same as 

 the angle between the other and PS. 



75. A particle describes an elliptic orbit about a focus ; prove that the 

 angular velocity at any point about the other focus varies inversely as the 

 square of the normal at the point. 



76. A particle describes any conic about a focus } prove that the total 

 velocity acquired in moving from one point to another is in the direction of 

 the line joining the focus to the pole of the chord joining the points. 



77. Prove that the periodic time of a particle projected with velocity V 

 from a point distant r from the origin, and having an acceleration ji/r 2 to 

 the origin, is 



2 

 f 



78. Prove that the greatest radial velocity of a particle describing an 

 ellipse about a focus is 



where *2a is the major axis, e the eccentricity, and jPthe periodic time. 



79. A particle describes an ellipse as a central orbit about a focus, and a 

 second particle describes the same ellipse in the same time with uniform 

 angular velocity about the same focus. The particles start together from the 

 farther apse. Prove that the angle the line joining the particles subtends at 

 the focus is greatest when the angle described by the first particle is 

 cos" 1 {1 - (1 -e 2 )?}/e, e being the eccentricity. 



80. A particle describes an ellipse of axes 2a, 26 about a focus. Prove 

 that the average distance of the particle from the focus for an indefinitely 

 great number of instants corresponding to equal differences of vectorial angle 

 is 6, and that the average distance of the particle from the focus for an 

 indefinitely great number of equidistant instants of time is a(l+|e 2 ), where 

 e is the eccentricity. 



81. When a parabola is described as a central orbit about a focus, prove 

 that the direction of motion at any point, P, meets the directrix in a point, Q, 

 whose velocity is inversely proportional to the abscissa of P. 



