180 MAXIMUM WORK DONE ST THE IMPULSE. 



101. Maximum Work done by the Impulse. 



The work done by the impulse P, from (4) of Art. 99, is 



W 

 Pv t = (1 - cos ) (v - t>,) v l (1) 



*/ 



The work is zero when the velocity of the surface v v = 0, 

 and also when it = v. 



To find the value of v l which makes this work a maxi- 

 mum, we must equate to zero its derivative with respect to 

 v lf which gives 



v %Vi = 0, or v l = %v; 



hence, the work done by the impulse is a maximum 

 when the surface moves in the direction of the stream, 

 with half the velocity of the stream. 



Substituting in (1) for v l its value, we have 



Pv, = (l_coB)i|jF f (2) 



which is the maximum work done by the impulse. 



COR. If the surface is a plane, as in Fig. 50, a = 90, 

 and we have, from (2), 



That is, the water transmits to the surface, in this 

 case, one-half of its kinetic energy. 



If the surface is hollow, as in Fig. 52, so that the water is 

 reversed, = 180, and we have, from (2), 



In this case, the water transmits to the surface all of 

 its kinetic energy. (See Weisbach's Mechs., p. 1010.) 



