77-82] THEORY OF MOMENTUM. 91 



The time-integral of a force between limits corresponding to 

 the beginning and end of any interval is called the impulse of the 

 force during the interval. It is the vector (unlocalised) whose 

 resolved part in any direction is the time-integral between the 

 same limits of the resolved part of the force in that direction. 



The above equations can be stated in words as follows : the 

 change of momentum of a particle in any interval is equal to the 

 impulse of the force acting on it during the interval. 



82. Impulsive action. Many changes of motion of natural 

 bodies take place with such rapidity that it is very difficult to 

 observe the gradual transition from one state of motion to another, 

 and it is therefore convenient, in our ideal system, to allow for the 

 possibility of sudden changes of motion. This is done by suppos 

 ing that the mutual action of two particles, or the resultant action 

 of many particles on one particle, can increase without limit as the 

 particle passes through some position. Let X lt Y lt Z 1 be the 

 sums of the resolved parts parallel to the axes of those forces 

 exerted on a particle m which do not remain finite throughout an 

 interval of time denoted by r, and suppose the instant at the 

 middle of this interval denoted by t, is the instant at which 

 X 1} YI, Z l cease to be finite. Then we suppose that 



Ct+^T rt+^r rt+^r 



Lt r= , Xdt, = X, Lt T=0 Y,dt, = F, Lt T = Q Z,dt, = Z, 



J t-fr J t-^r J t-\T 



are finite. We observe that these quantities are the limits of the 

 impulses of the forces during an indefinitely short interval con 

 taining the instant t. The vector, localised at the position of the 

 particle, whose resolved parts parallel to the axes are X, Y, Z is 

 defined to be the impulse exerted on the particle at the instant t. 



If the velocity of the particle just before the instant t has 

 resolved parts u , v 0) w parallel to the axes, and just after has 

 resolved parts u, v, w parallel to the axes, we have the equations 



mu mu = X, mv mv = Y, mw mw = Z y 



and these equations can be stated in words in the form : the 

 change of momentum of a particle in any direction is equal to the 

 impulse acting on the particle in that direction. These equations 

 are called equations of impulsive motion. 





