156 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



Further, when the orbit is an ellipse, its major axis, 2a, is the 

 sum of the greatest and least distances between the particles, and 

 the periodic time is by Article 53 equal to 



a* 



176. Examples. 



< 



1. If the particles are projected with velocities v, v' in directions con- 

 taining an angle a from points whose distance apart is R, prove that the 

 relative orbit is an ellipse, parabola, or hyperbola according as 

 _ 2^0' cos a<.= or 



2. S, P, and E denote the masses of the Sun, a planet, and the Earth ; the 

 major axis of the planet's orbit is Tc times that of the Earth's orbit, and its 

 periodic time is n years ; prove that 



[Kepler's Third Law of Planetary motion quoted in Article 94 states that 

 n 2 =k 3 approximately. This would follow if S were great compared with P or 

 K] 



3. Two gravitating spheres of masses m, m', and radii a, a', are allowed to 

 fall together from a position in which their centres are at a distance c, it is 

 required to find the time until they are in contact. 



We may suppose the centre of inertia at rest, and take x for the distance 

 between the centres of the spheres at time t. Then their velocities are 



m'x , mx 



, and ,. 

 m-fm m+m 



Hence the kinetic energy of the system is 



! , mm' . 



,v =ij -~ 2 



The potential energy, measured from the position in which the distance 

 was c as standard position, is (Article 142), 



, /I 1\ 



7mm' ) . 



\c x) 



Hence we have, by the equation of energy, 



and the time required is 



If then we find an angle such that a + a' = ccos 2 6, we shall have for the 

 required time 



