312 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



11. A particle is projected inside a smooth tube of equal mass which is 

 closed at both ends and lies on a smooth table. Prove that the distance 

 travelled through by the tube when the particle has made (n + l) impacts is 

 a(l-e n )/(e n -e n + 1 ) or a (I -e n + 1 )/(e n -e n + 1 ) according as n is odd or even, 

 2a being the length of the tube, and e the coefficient of restitution for each 

 impact. 



12. In a row of n equal spheres, the coefficient of restitution between 

 any two of which is e, one of the terminal spheres impinges directly on its 

 neighbour. Prove that their final velocities are in the ratios 



a : a/3 : a/3 2 : a/3 3 : ... a/3&quot;- 2 : $ n ~\ 

 where a = J(l-e) and j8=J(l +e). 



13. Two unequal particles are attached to a thread which passes over a 

 smooth pulley. Initially the smaller is in contact with a fixed horizontal 

 plane, and the other at a height h above the plane. Prove that, if the co 

 efficient of restitution for each impact is e, and if e is a root of any equation 

 of the form e n 2e 4- 1 = with n integral, the system will come to rest after 

 a time 2h(I + e)/v(l e) t where v is the velocity of the particle of greater 

 mass immediately before its first impact on the plane. 



14. Two equal spheres are in contact, and are attached by equal threads 

 to two other equal spheres at rest. The lines of the threads pass through 

 the centres of the spheres to which they are attached and make angles of 30 

 with that common tangent to the first two at their point of contact which 

 lies in the plane of the four centres. A fifth equal sphere running along 

 this common tangent strikes the first two symmetrically so that the threads 

 become tight. Prove that the velocity of the impinging sphere is diminished 

 in the ratio 7 12e : 19, where e is the coefficient of restitution. 



15. Two balls of masses M, m and of equal radii, connected by an 

 inextensible thread, lie on a smooth table with the thread straight, and a 

 ball of the same radius and of mass m moving parallel to the thread with 

 velocity v strikes the ball m so that the line of centres (m , m) makes an 

 acute angle a with the line of centres (M, m}. Prove that, if e is the co 

 efficient of restitution between m and m , M starts with velocity 



vmm (l+e) cos 2 a/{Mm sin 2 a + m (M+ m + m }}. 



16. Two balls are attached by inextensible threads to fixed points, and 

 one of them, of mass m describing a circle with velocity u impinges on the 

 other of mass m at rest, so that the line of centres makes an angle a with 

 the thread attached to m and the threads cross each other at right angles. 

 Prove that m will start to describe a circle with velocity 



mu sin a cos a(l+e)/(m cos 2 a-f m sin 2 a), 

 where e is the coefficient of restitution between the balls. 



