268 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



5. A particle is projected from the foot of a plane of inclination y in a 

 direction making an angle /3 with the normal to the plane, in a plane through 

 this normal making an angle a with the line of greatest slope on the inclined 

 plane. Prove that, for the particle to be on the horizontal through the point 

 of projection when it meets the plane for the nth time, the angles a, /3, y must 

 satisfy the equation 



( 1 - e n ] tan y = (1 - e) cos a tan /3. 



6. Three equal spheres are projected simultaneously from the corners of 

 an equilateral triangle with equal velocities towards the centre of the triangle, 

 and meet near the centre. Prove that they return to the corners with 

 velocities diminished in the ratio e : 1. 



7. A smooth uniform hemisphere of mass M is sliding with velocity V 

 on a plane with which its base is in contact ; a sphere of smaller mass m is 

 dropped vertically and strikes the hemisphere on the side towards which it is 

 moving so that the line joining their centres makes an angle rr/4 with the 

 vertical. Show that, if the coefficient of restitution between the plane and 

 the hemisphere is zero, and that between the sphere and the hemisphere is e, 

 the height through which the sphere must have fallen if the hemisphere is 

 stopped dead is 



8. A particle of mass M is moving on a smooth horizontal table with 

 uniform speed in a circle, being attached to the centre by an inextensible 

 thread, and strikes another particle of mass m at rest. Show that, if the two 

 particles adhere, the tension of the thread is diminished in the ratio 



M/(M+m). 



If there is restitution between the particles and the second one is describ 

 ing the same circle as the first, prove that the tensions T and t in the two 

 threads after impact are connected with their values before impact by the 

 equation 



T+ t = T, + t Q - (1 - e 2 ) y (m TJ - J(MtJ}*/(Ml m). 



9. A bucket and a counterpoise, of equal mass M, connected by a chain 

 of negligible mass passing over a smooth pulley, just balance each other, and 

 a ball, of mass m, is dropped into the centre of the bucket from a height k 

 above it ; find the time that elapses before the ball ceases to rebound, and 

 show that the whole distance descended by the bucket during this interval is 



10. Three equal particles are attached to the ends and middle point of a 

 rod of negligible mass, and one of the end ones is struck by a blow so that it 

 starts to move at right angles to the rod. Prove that the velocities of the 

 particles at starting are in the ratios, 5:2:1. 



11. An impulsive attraction acts between the centres of two spheres 

 which are approaching each other so as to generate kinetic energy E. If v is 



