125.] CENTRAL FORCES. 65 



ponents of its velocity v at the same time. The equation of the orbit 

 will assume the form 



for attraction, and 



*?(*oy -jo*) 2 - (x Q y -jo*) 2 = 



for repulsion. 



(2) Show that the semi-diameter conjugate to the initial, radius 

 vector has the length VQ/K, where v<? = x 2 + j> 2 . As any point of the 

 orbit can be regarded as initial point, it follows that the velocity at any 

 point is proportional to the parallel diameter of the orbit. 



(3) Find what the initial velocity must be to make the orbit a circle 

 in the case of attraction, and an equilateral hyperbola in the case of 

 repulsion. 



(4) The initial radius vector r and the initial velocity z> being given 

 geometrically, show how to construct the axes of the orbit described 

 under the action of a central force (of given intensity K 2 ) proportional 

 to the distance from the origin. 



(5) A particle describes an ellipse under the action of a central 

 force proportional to the distance ; show that the eccentric angle is 

 proportional to the time, and find the corresponding relation for a 

 hyperbolic orbit. 



(6) A particle of mass m describes a conic under the action of a 

 central force jp=^. m^r. Show that the sectorial velocity is \ c = 



r Kab, a and b being the semi-axes of the conic. 



(7) In Ex. (6) show that the time of revolution is T= 2?r/K, if the 

 conic is an ellipse. 



(8) A particle describes a conic under the action of a force whose 

 direction passes through the centre of the conic. Show that the force 



s proportional to the distance from the centre. 



(9) A particle is acted upon by two central forces of the same 

 ntensity (* 2 ), each proportional to the distance from a fixed centre. 



Determine the orbit : (a) when both forces are attractive ; (b) when 

 Doth are repulsive; (c) when one is an attraction, the other a re- 

 uulsion. 



PART in 5 



