IMP A CT A GAINST ANY SORFA CE OF KB VOL UTIO.V. 1 77 



w 



and in (7) gives P = 2v - (12) 



y 



COR. 4. When the surface is plane, as in Fig. 50, = 

 90 and cos = 0. Substituting this value in (5), (6), 

 and (7), they become 



W 

 /> = (?*)-, (13) 



which agrees with (6) of Art. 98 ; and 



W & 

 P = v = - aw, from (8), 



9 9 



= 2 x q- x aw = Zh x aw ; 



that is, the normal impulse of water against a plane 

 surface is equal to the weight of a column of water 

 whose base is equal to the cross-section of the stream, 

 and whose height is twice the head of water to which 

 the velocity is due. 



COR. 5. If the plane surface (Fig. 50) against which the 

 stream impinges moves away with a velocity in a direction 

 which makes an angle 9 with the original direction of the 

 stream, the velocity of the surface in the direction of the 



impact is 



v 1 = u cos 0, 



which in (13) gives for the impulse, 



P = (v u cos 0) , (15) 



J 



and the work done by it per second is 



W 



Pv v = (v u cos 6) u cos 6 . (16) 



J 



