2 KINETICS OF A PARTICLE. [3. 



hence, by integration, 



Jt- 

 Fdt=mv' mv, (2) 



where the time-integral in the left-hand member is the impulse 

 of the variable force ^during the time t' t. 



3. It appears, from equations (i) and (2), that a very large 

 force may produce a finite change of momentum in a very 

 short interval of time, but that it would require an infinite 

 force to produce an instantaneous change of momentum of 

 finite amount. The impact of one billiard ball on another, the 

 blow of a hammer, the stroke of the ram of a pile-driver, the 

 shock imparted by a falling body, by a projectile, by a railway 

 train in motion, by the explosion of the powder in a gun, are 

 familiar instances of large forces acting for only a very short 

 time and yet producing a very appreciable change of velocity. 

 The time of action, /' t, of such a force is the very brief period 

 during which the colliding bodies are in contact. The force, 

 F, is a pressure or an elastic stress exerted by either body on 

 the other during this time. 



Forces of this kind are called impulsive, or instantaneous, 

 forces. 



4. In the case of such impulsive forces, it is generally diffi- 

 cult or impossible by direct observation or experiment to deter- 

 mine separately the very brief time of action, t 1 t, as well as 



.the magnitude Fvi the impulsive force. Moreover, what is of 

 most practical importance and interest in such cases of impact 

 is, generally, not the force itself, but the change of momentum 

 produced, i.e. the impulse of the impulsive force. 



In the present section, which is devoted to the study of the 

 simplest cases of impact, we shall therefore deal with impulses 

 and momenta, and not with forces. 



5. It should be observed that many authors use the name impulsive, 

 or instantaneous, force for what has here been called the impulse of the 



