EXAMPLES 285 



20. A comet describes a parabola about the sun, its nearest distance 

 from the sun being one third of the radius of the earth's orbit, supposed 

 circular. For how many days will the comet remain within the earth's 

 orbit ? 



21. If the attraction on a particle varies as the inverse square of the 

 distance from a center of force 0, show that there are two directions in 

 which a particle can be projected from a given point P so that its orbit 

 may have a given major axis. If OP c, and if o^, a 2 are the angles 

 which the two directions of projection make with OP, show that 



where a is the semi major-axis. 



22. A particle is projected from a point P under a force to a fixed point 

 S at a distance R from P, so as to describe a circle passing through S. The 

 initial velocity is F, and the moment of the velocity about S is h. Show 

 that the particle will describe a semicircle in time 



JZ* 



4 h* 



23. A block of mass M, whose upper and lower faces are smooth horizon- 

 tal planes, is free to move along a groove in a parallel plane, and a particle 

 of mass m is attached to a fixed point in the upper face by an elastic string 

 whose natural length is a and modulus X. If the system starts from rest 

 with the particle on the upper face, and the string stretched parallel to 

 the groove to 1-f n times its natural length, prove that the block will per- 

 form oscillations of amplitude 



(n + 1) am 



M+m 

 and period 



aMm 



