Setting up First Order Differential Equations
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1. In the following, represent the situation pictorially and deduce the governing 1st order differential equations and initial condition for the quantity in question.
a) A tank, containing 300 gallons of pure water initially, is emptied out in the following fashion. A salt solution of concentration ½ lb of salt per gallon is allowed to enter the tank at a constant rate and the well stirred mixture is emptied at twice that rate. The amount of time it takes to drain the tank completely is 60 minutes. The quantity in question is the amount of salt in the tank at any time before it runs dry.
b) A trapped styrofoam sphere of radius "a" and density rho_0 is released from a submerged object and rises vertically from rest in a fluid of density rho_1 > rho_0 . Assume that the fluid medium offers resistance to this sphere proportional to the instantaneous speed (magnitude of the velocity) of the latter and acting in a direction to oppose motion. Further consider the acceleration due to gravity to be a constant, g, and the proportionally constant of resistance equal to k>0. The quantity in question is the velocity of rising sphere as a function of time.
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Solution Summary
This solution provides an example of how to convert a word problem into a differential equation with an initial condition. Two examples are shown: the first is a mass balance on a vat of salt water with inlet and outlet; the second is a motion problem (kinematics) involving buoyancy. Solution includes hand-drawn diagrams and detailed explanations in pdf file format.
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