EXAMPLES. 167 



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

 followed, at a distance r, by a smaller body of mass ra 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 \ TT + V(l - w 2 ) + cos 

 \I+w) 



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 



2 tan ~ * [y (m + m')/ V 2 d}. 



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 o>, 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 < ; obtain the equation 



2w - raxf> = 3o> (r<j> -f 2om) + const., 

 squares of u and <p being neglected. 



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

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

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



