252 MOTION OF SYSTEMS OF PARTICLES 



12. In the system of pulleys described in 128, show that if P is a weight 

 which is not equal to W/n, the acceleration produced in the weight W will be 



nP-W 

 n*P+W 9 ' 



13. Two masses m, m' connected by an elastic string are placed on a 

 smooth horizontal table, the masses being at rest and the string unstretched. 

 A blow of impulse P is given to the first mass, in the direction away from 

 the second. Show that when the string is again unstretched, the velocity 

 of the second mass is 



m + m' 



14. Three equal particles are tied at the ends and middle point of an 

 inextensible string, which is placed, fully extended, on a smooth table. 

 The middle particle is jerked into motion in the direction towards and 

 perpendicular to the line joining the other two. Find the loss of energy 

 when the other particles are jerked into motion. 



15. A coal train consists of a number of similar trucks hauled by an 

 engine whose weight is just equal to that of three trucks. The train is at 

 rest on a level track, the couplings, which are of equal length, being al] 

 equally slack. The engine then begins to move with a constant tractive 

 force, and each truck is jerked into motion as its coupling tightens. Show 

 that the speed of the engine will be greatest just before the tenth jerk 

 occurs. 



16. Snow is evenly spread over a roof. If a mass commences to slide, 

 clearing away a path of uniform breadth as it goes, prove that its acceler- 

 ation is constant, and equal to a third of that of a mass sliding freely 

 down the roof. 



17. A heavy, perfectly flexible uniform string hanging vertically with 

 its lowest point at a height h above an inelastic horizontal plane is sud- 

 denly allowed to fall on to the plane. Show that the pressure on the table 

 when a length x of the string has fallen on to the table is 



(3 x + 2 7i) mg. 



18. Show that if two equal balls impinge directly with velocities 

 e V and V, the former will be reduced to rest. 



1 



19. Show that the mass m of a sphere which must be interposed between 

 a sphere of mass M at rest and one of mass M' moving directly on to it 

 with velocity V, in order that the former may acquire the greatest possible 

 velocity from the impact, will be VjOf 7 , and that the velocity acquired 



e M'V(\+ef 



M+M'+2m 



