EXAMPLES. 311 



4. Two spheres of equal radius and of masses X^/n and X 2 m are lying in 

 contact on a smooth horizontal plane. A third sphere of the same radius 

 and of mass m falls freely with its centre in the vertical plane containing 

 the centres of the other two so as to strike them simultaneously. Assuming 

 that there is no restitution in any of the impacts prove that the velocity 

 produced in the sphere of mass X x m is 



yV3(l + 2X 2 )/(l + 4^ + 4X3 + 12X^2), 

 where v is the velocity of the falling sphere just before impact. 



5. From one corner A of a rectangular billiard table a ball is projected 

 in a direction making an angle a with the side AB ; it strikes first the side 

 Z?(7, then AD, then DC, then EC again, and then returns to A. Prove that, 

 if e is the coefficient of restitution, AB : AD = e 2 cot a : l + e 2 . 



6. Three smooth billiard balls of perfect restitution, each of radius d, 

 rest on a smooth table, their centres forming a triangle ABC ; prove that, 

 if the ball A is to cannon off B on o (7, the angle of impact at B must lie 

 between 



D . . , 



B- |TT -tan- 1 - -^r^-^ and B + d-%7r -tan" 

 c-2dsmB 



where 8 = sm~ 1 4d/a. 



7. Two bodies of masses m, m', (m>m f ), are connected by a string 

 passing over a smooth pulley, the coefficient of restitution between the 

 greater and the plane being unity and that between the smaller and the 

 obstacle zero. They start from rest at the same distance a above a fixed 

 horizontal plane, and, when m impinges on the plane, m' strikes against a 

 fixed obstacle. Show that the two bodies are again in a position of in- 

 stantaneous rest when m is at a height m?a/(m + m') 2 above the plane. 



8. Show that it is possible to project a small elastic ball inside a regular 

 polygon of n sides so as to describe a regular polygon of the same number 

 of sides, and prove that the ratio of the sides of the two polygons is 



7T f // 1-6 . 27T\ //, 1-6 . 27T\1 



2 cos - : -! / ( 1 + _ - sin 1 + / 1 - - sin H , 

 n lv \ l + / V \ l + e n J) 



where e is the coefficient of restitution. 



9. A ball is projected with velocity V from a point of a plane inclined 

 at an angle a to the horizontal, the direction of projection is at right angles to 

 the plane, and the coefficient of restitution between the ball and the plane 

 is e. Prove that, before ceasing to bound, it will have described a length 

 2 V 2 sin a/<7 cos 2 a (1 - e) 2 along the plane. 



10. A hollow elliptic cylinder stands on a horizontal plane with its axis 

 vertical. From the focus of a horizontal section a particle is projected in a 

 horizontal direction with velocity v. Prove that if it returns to the point of 

 projection the height of the section above the table is 2m 2 ^a 2 /n 2 v 2 , where m, n 

 are any integers and 2a is the major axis, the coefficient of restitution in 

 each impact being unity. 



