EXAMPLES. 167 



35. A body of mass M is moving in a straight line with velocity 7&quot;, and is 

 followed, at a distance r, by a smaller body of mass m moving in the same line 

 with velocity u. The bodies attract each other according to the law of gravi 

 tation. Prove that the smaller body will overtake the other after a time 



/ r \ rr + V(l - H&amp;gt; 2 ) + cos&quot; 1 w 

 \l+w) ~ 



r-u 

 where l-w = -~~ . 



y (M+ m) 



36. Two bodies, masses m, m , are describing relatively to each other 

 circular orbits under their mutual gravitation, a and a being their distances 

 from the centre of inertia. If F is the relative velocity, and m receives an 

 impulse mV towards m , prove that the two bodies proceed to describe, 

 relatively to the centre of inertia, parabolas whose latera recta are 2a and 2a . 



37. Two gravitating spheres of masses m, m moving freely have relative 

 velocity V when at a great distance apart, and in the absence of gravitation 

 one would pass the other at a minimum distance d. Prove that the relative 

 orbits are hyperbolic, and that the direction of the relative velocity will be 

 ultimately turned through an angle 



38. In a system of two gravitating bodies, M and m, initially M is at rest, 

 and m is projected with velocity J{y(M+m)ld} at right angles to the line 

 joining the bodies, d being the distance between the bodies. Prove that the 

 path of M is a succession of cycloids and that M comes to rest at a cusp after 

 equal intervals of time. 



39. In a system of two gravitating bodies of masses M and m the relative 

 orbit is an ellipse of semi-axes a and b. Prove that if the mass of the second 

 body could be suddenly doubled the eccentricity of the new orbit would be 



where v is the relative velocity at the instant of the change. 



40. Two gravitating particles whose distance is r, are describing circles 

 uniformly about their common centre of gravity with angular velocity &amp;lt;&amp;gt;, and 

 a small general disturbance in the plane of motion is communicated to the 

 system, so that after any time t the distance is r + u, and the line joining the 

 particles is in advance of the position it would have occupied if the steady 

 motion had not been disturbed by the angle &amp;lt; ; obtain the equation 



2u - ra&amp;gt;(f) = Sat (rj&amp;gt; + 2a&amp;gt;u} + const., 

 squares of u and being neglected. 



41. Two equal particles P, Q are projected from points equidistant on 

 opposite sides of a third particle $, with a velocity due to their distance under 

 the attraction of S only. All three particles are gravitating, and the directions 



