10 KINETICS OF A PARTICLE. [21. 



last ball of the row will move off with the velocity u, while all the other 

 balls will remain at rest. 



(10) Find the velocity of the last (th) ball in Ex. (9), when the 

 coefficient of restitution is e. 



(n) An inelastic ball of 8 Ibs. is moving with a velocity of 12 ft. per 

 second, (a) With what velocity must a ball of 24 Ibs. meet it to arrest 

 its motion? () With what velocity would the ball of 24 Ibs. have to 

 impinge from behind on the ball of 8 Ibs. to double its velocity ? 



(12) A ball m impinges upon a ball m' from behind with a velocity 

 u. Determine the velocities after impact, both for inelastic and for per- 

 fectly elastic balls : (a) when -m' is originally at rest; (b) when m' is 

 at rest and very large in comparison with m ; (<r) when m 1 has the 

 initial velocity z/', and is very large in comparison with m. 



21. Kinetic Energy. A particle of mass m t moving with the 

 velocity v, has the kinetic energy ^mv 2 (Part II., Art. 71). As 

 this is not a vector-quantity, the kinetic energy of a system 

 consisting of any number of free particles is simply the alge- 

 braic sum, ^^mv^y of the kinetic energies of these particles. 

 It is an essentially positive quantity, provided the masses are 

 all positive. 



The kinetic energy of a rigid body having a motion of pure 

 translation is evidently =\mv i y if m be the mass of the body 

 and v the common velocity of all its points. 



22. Change of kinetic energy is brought about by the action 

 of force, and we have (Part II., Art. 72) for a constant force F t 



\mv'*-\mv*=F(s>-s); (u) 



and for a variable force F, 



' (12) 



where the quantity in the right-hand member is called the work 

 of the force. Thus a particle, of mass m, falling from rest 

 through a distance s, acquires its kinetic energy owing to the 



