p0650 - 6.50 CHAPTER 6 PROBLEM 50 599 6.50 Chapter 6...

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6.50. CHAPTER 6, PROBLEM 50 599 6.50 Chapter 6, Problem 50 We select a control volume translating to the right with velocity u cv = U i , using the boundaries indicated in Figure 6.50. Figure 6.50: Cart propelled by a water jet—control volume moving. Mass conservation: In general, the mass-conservation principle tells us that d dt 888 V ρ dV + 8 s 8 S ρ u rel · n dS = 0 The flow is incompressible and the control volume’s size is constant. Thus, the instanta- neous rate of change of the mass in the control volume vanishes. Then, since u rel = u U i , we have 8 s 8 S ρ ( u U i ) · n dS = 0 The incident jet velocity is u = U j i . Let v and A denote the magnitude of the outflow velocity relative to the cart and area in each jet at the sides of the cart. Then, ρ ( U j i U i ) · ( i ) π 4 d 2 ± 1 Incident jet + ρ ( v j ) · j A ± 1 Top jet + ρ ( v j ) · ( j ) A ± 1 Bottom jet = 0 Simplifying, we can solve for vA to discover that vA = π 8 ( U j U ) d 2 x -Momentum Conservation:
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