THE HYDRAULIC BRAKE 



791 



Brake with Passages of Constant Area. In this case we have, neglecting 

 mechanical friction, 



A* 



, T , 2 , 62-4 L fl] A 3 



= k V 2 where k = ~~ \ 1 + [ $ 



Since this measures the mass X acceleration of the moving body 



(f d t g d x 



the negative sign indicating that the motion is being retarded. 



or 



g dx 



~*k 



The solution of this equation gives : 



Writing x 1 = 0, so that FI is the velocity at the instant of impact, we 

 have 



F kg 



Io &Fi=--Tf*' 



~w x > 

 e 



or F = 



giving the velocity corresponding to any position x of the piston. Since 



~~W Xt 

 e never becomes equal to zero, no matter how large the value of x, 



the velocity cannot become zero until a change takes place in the law of 

 resistance, either on account of the effect of solid friction, or because of 

 the motion of the fluid becoming so slow that the resistance becomes 

 approximately proportional to the first power of the velocity. 



On taking mechanical friction into account we have, assuming this to 

 be constant, 



dx 



or 



+ b I'' = c, 





 + ^ 



where 



, 2 9 F 

 W 



32-4 A* 

 W a 2 



fj\ 

 ml* 



1 2 q 1 



( c = --w 



