38.] RECTILINEAR MOTION. X 9 



ponents d z x/dfi, d^y/dt 2 , d 2 z/dfi, and .Finto the components X, 



' ' 



37. To familiarize the student with the idea of force and its 

 use in kinetics, we shall now study in some detail the simple 

 case of rectilinear motion. The next section will be devoted to 

 the general problem of the curvilinear motion of a. free particle. 

 This will be illustrated by the important case of motion due 

 to central forces. Finally, the motion of a particle subject to 

 conditions, or constraints, will be treated. 



38. When a particle of mass m moves in a straight line, both 

 its velocity v and the resultant force F must be directed along 

 this line. The acceleration in rectilinear motion (see Part I., 

 Art. 103) is j = dv / dt = d*s / dt* ', hence the dynamical equation 

 of rectilinear motion y 



-%--%-* ^ > 



It differs from the kinematical equation (Part I., Art. 1 1 5) only 

 by the factor m, and can be treated in the same way. 



Thus, if the law of force be given, i.e. if F be known as a 

 function of t, s, v, or of only one or two of these quantities, the 

 equation can be integrated ; and if, moreover, the initial position 

 and velocity of the particle be given, the constants of integra- 

 tion can be determined, and all the circumstances of the motion 

 can be found. 



If the mass m of the moving particle were not a constant 

 quantity, the equation (i) should be written in the form 



d(mv) _ p 



~dT *' 



since the resultant force is the rate at which the momentum of 

 the particle changes with the time (see Part II., Art. 60). 



