CHAPTER XIII. 



REVERSAL OF THE PROBLEM OF KINEMATIC PROGNOSIS. KINEMATIC 

 DETERMINATION OF ACCELERATION. 



194. On the Reversed Problems. According to our general plan (section 96) 

 we shall now consider the problem of kinematic prognosis in its reverse form. 

 Knowing from the observations the initial and the final state of motion, we shall 

 investigate the change of motion which has led from one state to the other. This 

 will involve a determination, on pure kinematic principles, of the acceleration of 

 the moving particles. 



If we ever succeed in giving the complete solution of the problem of prognosis, 

 we shall have to determine the accelerations not by kinematic but by dynamic 

 methods. This should be theoretically possible because the observations should 

 allow us to derive the forces which produce the accelerations. But passing to 

 the practical performance, we shall meet with a great difficulty. Though we know 

 from laboratory experiments the coefficient of the friction of the air, we shall not 

 be able to use it practically for determining the influence of friction on acceleration. 

 The reason is obvious. Friction depends upon true motion, while we are forced to 

 work with an idealized motion, disregarding all the small irregularities of the motion 

 rsection 97) and of the ground (section 179). If we were to determine the frictional 

 resistance in the rational way we should have to examine the motion from milli- 

 meter to millimeter, and not only at stations which may be hundreds of kilometers 

 from each other. As this will not be possible, we shall be obliged to find other ways 

 for determining the influence which, as an indirect effect of friction, modifies the 

 idealized motion which we consider. We must develop methods for determining, 

 by pure kinematic principles, the acceleration of the idealized motion to the consider- 

 ation of which we are confined, and by comparing it with the accelerating forces 

 find empirical rules for taking the effect of friction into account. 



As an introductory problem to the kinematic determination of accelerations, 

 we shall first treat the problem of the identification of particles on two successive 

 charts of motion or (what comes to the same thing) the determination in the 

 second approximation of the displacement of these particles. 



195. Determination of Displacements in the Second Approximation. Let a 

 chart be given which represents the state of motion at the epoch t . We shall 

 consider a particle which at this epoch is situated at the point A (fig. 112). Accord- 

 ing to the chart it has a certain velocity v . During the short interval of time t l / 

 its displacement will then in the first approximation be 



(a) AB' = v (t x -Q 



Thus the point B' will give in the first approximation the situation of the particle 



at the epoch/,. 



'63 



