IMP A CT A G AINST A N Y SURFA CE OF RE VOL UTION, 175 



V = V(V ,)> + t>j* +~ 2 (V Vj) V t COS C(. (I) 



Now the kinetic energy of the water before impact 



V 3 



= f W > 

 ty 



and the kinetic energy remaining in the water after impact 



F 2 



= **> < 2 > 



hence, the kinetic energy transmitted to the surface 



W 



= ( ' ;2 - F2) lr < 3) 



If P be put for the force or impulse against the surface, 

 then the energy transmitted to the surface = Pv iy which in 

 (3) gives 



- [y2 _ ( V - Vl )2 _ Vl 2 _ 2 (V - Vj V^ COS ] ^, 



from (1), 



W 

 = (1 - cos ) (o vj v l ; (4) 



/. P=(l-cosa)(v-v 1 )^, (5) 



which is the force of the water against the surface in the 

 direction of the axis. 



That is, the impulse varies as the relative velocity of 

 the water. 



COR. 1. If the surface moves with a velocity v l in the 

 opposite direction to that of the water, we have, from (5), 



W 



p = (i _ C os)(y + r,) (6) 



7 



