54 THE MOVING HYDRAULIC JUMP 



The other way to apply the law of continuity is as follows: Move 

 the shaded area of Fig. 501 (c) to the left an amount equal to Vodi, the 

 distance that the jump has traveled, and then superpose this area upon 

 the area shown in Fig. 501 (b) so that ek will coincide with dc, as shown 

 in Fig. 501 (J). Then bh will likewise have moved to the left a dis- 

 tance Vodt to b'h', and mn to m'n. The two shaded areas as shown 

 in Fig. 501 (<i) will only partially coincide, leaving at the left the area 

 ab'h'j and at the right fm'ng, which by the law of continuity must be 

 equal. Therefore, 



D,{Vi- Vo) = D2{V2- Vo) [502] 



which is the equivalent of equation (501) by a simple algebraic trans- 

 position. 



To apply the law of change of momentum, the difference between the 

 momentum of the mass of water shown in Fig. 501 (&) and the mo- 

 mentum of the same mass of water in its later position Fig. 501 (c) must 

 be found. The momentum of a mass is found by taking the sum of the 

 momenta of its parts. When the two masses are superposed as in Fig. 

 501 (d) the portion between b'h^ and fg is common to both masses. 

 Therefore, the desired change in momentum is the momentum of the 

 mass ab'h'j minus the momentum of the equal mass fm'ng, or it is the 

 mass ab'h'j multiplied by the change in velocity V\ — ¥2- 



The momentum of any body can be changed only as the result of an 

 external unbalanced force acting upon the body. The momentum of 

 the mass of water shown in Fig. 501(c) is less than the momentum of 

 the same mass in the position Fig. 501(6), because the hydrostatic 

 force acting towards the left against the face fg is greater than the 

 hydrostatic pressure acting towards the right against the face aj. The 

 difference between these two pressures is the unbalanced force which 

 reduces the momentum of the mass of moving water while it travels 

 from position Fig. 501 (&) to position Fig. 501(c). The pressure 



against the face fg is - Dz^, against the face aj, — Di^. 



The law of the change of momentum is that the unbalanced force is 

 equal to the time rate of change of momentum. In this case the 

 weight of water to be considered is the weight of ab h j, equal to 

 wDiiVi — Vo)dt, the change in velocity is Vi — V2, and the time 

 during which the change takes place is dt. 



Therefore, 



^^^2 n2^ wDiiVi - Vo)dt{Vi - V2) 



{D2' - D,') = 



gdt 



