90 MASS AND FORCE. [CHAP. V. 



When we speak of the acceleration of a body we must be 

 understood to mean the acceleration of the centre of inertia of the 

 body supposed to be rigid.' 



78. Momentum. The momentum of a particle is defined to 

 be a vector localised in the line of its velocity, having the sense 

 of the velocity, and of magnitude equal to the product of the mass 

 of the particle and its velocity. 



79. Kinetic Reaction. The kinetic reaction of a particle 

 is a vector localised in the line of its acceleration, having the 

 sense of the acceleration, and of magnitude equal to the product 

 of the mass of the particle and its acceleration. 



The magnitude of this vector, its sense, and the line in which 

 it is localised are the same as those of the rate of change of 

 momentum of the particle per unit of time. 



80. Equations of motion. Since the acceleration of a 

 particle is the resultant of the accelerations contributed to it by 

 the actions of other particles, its kinetic reaction is the resultant 

 of the kinetic reactions produced by these particles ; these com- 

 ponent kinetic reactions are vectors identical in magnitude and 

 sense with the forces acting on the particle, and are localised in 

 the lines of action of these forces. It follows that the resolved 

 part, parallel to any line, of the kinetic reaction of the particle is 

 the sum of the resolved parts, parallel to the same line, of the 

 forces that act on the particle. 



The equations expressing this equivalence are called the 

 equations of motion of the particle. 



Thus let x, y, z be the coordinates of the particle at time t, m 

 its mass, and let X, F, Z be the resolved parts parallel to the 

 axes of the resultant force acting upon it. The equations of 

 motion are 



mx = X, my = F, mz = Z. 



81. Impulse of a force. Let 0) y Q) z be the resolved 

 parts of the velocity at the instant when t = t Q . On integrating 

 the above equations we have 



rt 



mx m# = I Xdt and two similar equations. 

 J tit 



