7, 8, 9] 



SECTOR VELOCITY. ACCELERATION. 



13 



9. Acceleration. If the vetycity of a point is variable with the 



time we define the acceleration of the point as the limit of the ratio 



of the increment of velocity At? to the increment of time A, as 



both approach zero. We may consider either the numerical change 



A vi dv d^ s 



r?nAt = d* = d?' - C 



or the geometrical change. 

 If we draw a vector AB 

 (Fig. 6) to represent the velo- 

 city at the time t and the 

 vector AC to represent the 

 velocity at the time t -f A t, 

 and draw the arc of a circle 

 BD, DC will represent the 

 numerical change of velo- 

 city, Av ; not considering its 

 direction, while BC re- 

 presents its geometrical, or vector change, 



Fig. 6. 



for 



= AC-AB = 



Accordingly lim -r = lim -^7 is the vector acceleration a . 

 Ji=o &t Jt=o A * 



Since the projections of the geometrical difference of two vectors 

 are the differences of the projections, the components of H in any 

 direction will be proportional to the changes of the corresponding 

 components of the velocities, that is 



26) 



d *y 

 dt 



dv 2 

 ~dt 



It* 



dt 



dt 



In the language of 6, the acceleration is the velocity of the 

 velocity -vector. 



The vector acceleration a being the resultant of the components 

 a*, a y , a z , has the numerical value or tensor 



