By Peter V. O'Neil
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Think, for example, of a brick dropped in a swimming pool or a ball bearing dropped in a tan k of oil . We, want to analyze the object's motion . _ Let v(t) be the velocity at time t . The force of gravity pulls the object down and ha s magnitude nag . The medium retards the motion. The magnitude of this retarding force is not obvious, but experiment has shown that its magnitude is proportional to the square o f the velocity . If we choose downward as the positive direction and upward as negative, the n Newton's law tells us that, for some constant a, dv dt If we assume that the object begins its motion from rest (dropped, not thrown) and if we star t the clock at this instant, then v(0) = O .
CHAPTER 1 First-Order Differential Equations 26 PROBLEMS 16. A 500-gallon tank initially contains 50 gallons of brin e solution in which 28 pounds of salt have been dissolved . Beginning at time zero, brine containing 2 pounds of salt per gallon is added at the rate of 3 gallons per minute, and the mixture is poured out of the tank a t the rate of 2 gallons per minute . How much salt i s in the tank when it contains 100 gallons of brine ? Hint : The amount of brine in the tank at time t is 50 + t .
Suppose we make the second measurement at that time T = H at which exactly half of the mass has radiated away . At this time, half of the mass remains, s o MT = M/2 and MT /M = 1/2. Now the expression for the mass become s m(t) = Me ln(1/2)t/ x or 7 2 (t) = Me- in (2)t/H This number H is called the half-life of the element . Although we took it to be the tim e needed for half of the original amount M to decay, in fact, between any times t 1 and t 1 +H , exactly half of the mass of the element present at t l will radiate away .
Advanced Engineering Mathematics by Peter V. O'Neil