Problem 4.151 - Problem 4.151[Difficulty 4 Given Vaned cart...

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Problem 4.151 [Difficulty: 4] Given: Vaned cart being hit by jet Find: Jet speed to stop cart in 1s and 2 s; distance traveled Solution: Apply x momentum Assumptions: 1) All changes wrt CV 2) Incompressible flow 3) Atmospheric pressure in jet 4) Uniform flow 5) Constant jet area Given data ρ 999 kg m 3 = M 5 kg = D 35 mm = θ 60 deg = U 0 5 m s = A π 4 D 2 = A 962 mm 2 = Then a rf M u 1 ρ V U + ( ) A [ ] u 2 ρ V U + ( ) A [ ] + = where a rf dU dt = u 1 V U + ( ) = and u 2 V U + ( ) cos θ ( ) = Hence dU dt M ρ V U + ( ) 2 A ρ V U + ( ) 2 A cos θ ( ) = ρ V U + ( ) 2 A 1 cos θ ( ) ( ) = or dU dt M ρ V U + ( ) 2 A 1 cos θ ( ) ( ) = (1) d V U + ( ) V U + ( ) 2 ρ 1 cos θ ( ) ( ) A M dt = Note that V is constant, so dU = d(V+U), separating variables Integrating from U 0 at t = 0 to U = 0 at t 1 V 1 V U 0 + ρ 1 cos θ ( ) ( ) A M t = Solving for V U 0 V V U 0 + ( ) ρ 1 cos θ ( ) ( ) A t M = or V 2 V U 0 + M U 0 ρ 1 cos θ ( ) ( ) A t Hence V U 0 2 U 0 2 4 U 0
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