Nearly frictionless vertical guide rails maintain a plate of mass mp in a horizontal position, such that it can slide freely in the vertical direction. A nozzle directs a water stream of area A against the plate underside. The water jet splatters in the plate plane, applying an upward force against the plate. The water flow rate ṁ (kg/s) can be controlled. Assume that distances are short, so the velocity of the rising jet can be considered constant with height.

(a) Determine the minimum mass flow rate ṁ_min necessary to just levitate the plate and obtain a relation for the steady-state velocity of the upward moving plate for ṁ > ṁ_min.

(b) At time t = 0, the plate is at rest, and the water jet with ṁ > ṁ_min is suddenly turned on. Apply a force balance to the plate and obtain the integral that relates velocity to time (do not solve).

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Guide rails mp Nozzle