For a structure or any other large object floating in the water, the 

 mass, m, of the object is equal to the displaced mass, pV, of the 

 water. This mass may vary as water gradually floods the interior of a 

 structure, but for the analysis presented here the mass will be assumed 

 constant. From Newton's second law 



du, du, 



F = m - — = pV 



dt dt 



(338) 



At any instant in time the magnitude of the deceleration of the fluid 

 with respect to the object is equal to the magnitude of the acceleration 

 of the ground with respect to the object (which is equal to the accelera- 

 tion of the object with respect to the ground), i.e., where u is assumed 

 constant, 



so equation (337) becomes 



du 



F = pV 



ft 



dt 



d(u 



- V 



= _^ 









( 



it 





dt 









~ C D 



pA 



(u 



-v 2 



C„ 



M 



PV 



du 





2 



dt 



(339) 



(340) 



or, rearranging terms, 



: zA 



\ 



dt 2v(i + cy 



(u - up' 



(341) 



For an object moving a short distance, the coefficients Cp and Cy 

 will be assumed constant. This is not entirely correct (e.g., the value 

 of Cj) will vary as a function of velocity), but will be assumed as 

 approximately correct for a short distance. A constant, a, can then 

 be defined by 



CpA 



2v(i * cy 



(342) 



Substituting equation (342) into equation (341) and rearranging terms give 



a dt 



Integrating equation (343) , 



at 



(u-up 



r u b _ 



"Jo (u 



\ 



V 2 u " "* u 



(343) 



(344) 



186 



