Q, F. Becker — Impact Friction and Faulting. 119 



From the instant at which an impinging mass first comes in 

 contact with a passive mass, to the moment at which the cen- 

 ters of inertia come to rest relatively to the center of inertia of 

 the system, the active and reactionary forces are in equilibrium. 

 This is manifestly the case for elastic bodies, and since during 

 this period elastic and inelastic bodies behave exactly alike, 

 inelastic bodies are reducible to a conservative system for the 

 same period. The principle of virtual velocities is therefore con- 

 tinuously applicable and the energy potentialized (or expended) 

 in the two masses is equal in absolute value and opposite in 

 direction. The kinetic energy at the moment of maximum 

 compression on the other hand will be uniformly distributed 

 (relatively to the mass) over the entire system. 



Solution for finite compressible rod. — The geometrical methods 

 of representing energy are as various as the corresponding 

 algebraic notations, but perhaps the most natural is that in 

 which the energy of a moving body is made proportional to 

 the volume of the body and to its energy per unit of volume. 

 If the energy potentialized throughout a given volume were 

 uniformly distributed, the quantity of energy potentialized in a 

 cubic unit would then be, say w, and that in an infinitesimal 

 cube would be todxdydz. Suppose a finite compressible cylin- 

 der at rest to suffer impact from a second similar cylinder 

 moving with a velocity v. It will be convenient to consider 

 the mass of the moving cylinder as 2M and that of the pas- 

 sive cylinder as 2nM. The entire energy potentialized in -the 

 passive mass at the moment of maximum compression will then 

 TVf ?; 2 n 



be — z, , and the kinetic energy of the whole system at the 



2 tt+1 J 



2Mv 2 1 

 same instant will be — ^ — :. The problem proposed is to 



find an expression for the distribution of potentialized energy 



throughout the passive mass, or to state w in terms of x, for 



the moment of maximum compression. 



Between any two successive sections w will be diminished 



by two quantities, one representing the kinetic energy imparted 



to the mass between these sections, and the other the energy 



potentialized. Indeed these quantities may be thought of 



separately as if a certain amount of kinetic energy were first 



distributed uniformly over the passive mass and afterward a 



certain quantity of internal work were done in it. If the 



length of the entire system is unity, the diminution of w betwen 



x and x+dx due to the uniform distribution of kinetic energy 



2Mv 2 dx 



will be — K r . The energy potentialized between these 



2 n + 1 OJ r 



limits is of course todx, but this quantity does not bear a simple 



relation to dw unless none of the energy assumes the kinetic 



