;.] IMPULSES. 3 



impulsive force. They define an impulsive force as the limiting value 



Jt r 

 Fdt when F increases indefinitely, while at the same 



time the difference of the limits, / /, is indefinitely diminished ; in 

 other words, as the impulse of an infinite force producing a finite 

 change of momentum in an infinitesimal time. 



According to this definition, an impulsive or instantaneous force is a 

 magnitude of a character different from that of an ordinary force, and is 

 measured by a different unit. Its dimensions are MLT" 1 , and not 

 MLT~ 2 . Its unit is the same as the unit of momentum. Indeed, it is 

 not a force, but an impulse. 



We can arrive at this idea of an instantaneous force from a some- 

 what different point of view. Just as in kinematics (Part I., Arts. 104 

 and 156) we may distinguish accelerations of different orders, regard- 

 ing velocity as acceleration of order zero, so in dynamics instantaneous 

 forces may be regarded as forces of order o, ordinary (continuous) 

 forces as forces of order i, the product of mass into the acceleration of 

 the second order as a force of the second order, and so on. 



In the present elementary treatise, no use is made of these considera- 

 tions. The word force is always used as meaning the product of mass 



C* 



into acceleration of the first order, and the time-integral I Fdt is always 



called impulse, and not impulsive force. 



6. The momentum mv of a particle P of mass m, moving 

 with the velocity v, can be represented geometrically by a 

 vector (more exactly by a localized vector, or rotor), i.e. by a 

 segment of a straight line drawn through P and representing 

 by its length the magnitude of the momentum, by its direction 

 and sense the direction and sense of the velocity. Hence, the 

 composition and resolution of momenta follows the same rules that 

 hold for forces. 



7. Let us consider two particles P lt P^ of masses m^ m^ 

 having equal and parallel velocities v, and let their momenta. 

 m-p, m^v be represented by their vectors (Fig. i). The two 

 particles may be regarded as forming a single moving system ; 

 as the velocities are equal in magnitude, direction, and sense, 

 the system has a motion of translation. According to the rule 



