I4I-] 



VELOCITY. 



139. The vector PTma.y also be regarded as the limit of a vector 

 PS laid off on the chord PP as before, but proportional to the velocity 

 with which the point would describe the chord PP in the time A/, i.e. 



to the velocity PS= For as A/ approaches the limit o, 



A/ 



PS approaches the direction of the tangent, and the ratio of the arc 

 AJ to the chord PP approaches the limit i. Hence the equation 



= - - PS gives in the limit lim = lim PS t or PT lim PS. 

 A/ chord PP A/ 



It may be noticed here that, in view of the practical applications, the 

 function /(/) = s is in mechanics always supposed to be itself continuous 

 and to possess continuous and finite derivatives of the first and second 

 order. 



140. Velocity having thus been denned as a vector, we may 

 at once apply to it the rules for vector composition and vector 

 resolution laid down in Arts. 45-55 for vectors representing dis- 

 placements. Thus if a point be subjected to two or more 

 simultaneous velocities, the velocity of the resulting motion will 

 be represented by the vector found by geometrically adding the 

 component velocities. A velocity may be resolved into any 

 number of component velocities whose geometrical sum is equal 

 to the given velocity. 



141. We proceed to consider the most important cases of 

 resolution of a velocity in a 



plane. 



Let a point P move in a 

 curve P Q P (Fig. 33) whose 

 equation is referred to rec- 

 tangular Cartesian co-ordi- 

 nates x y y. It is usually con- 

 venient in this case to 

 resolve the velocity v par- ' 



into v x 



Fig. 33. 



allel to the axes 

 and v y . 



If a be the angle made by the vector v with the axis of x, we 

 have v x = vcQsa, v, = vsina. And as the element ds of the 

 curve at P makes the same angle a with the axis of x, we also 



