Differential Equations Problem Set
Solve the following problems with Maple, then compare your answers to my solutions. See Kinematics 1 Problem Set for instructions on how to avoid switching back and forth between a problem statement in your browser window and your Maple worksheet.
In all of these problems assume that the acceleration due to gravity is 10 m/s^2.
(1) Find the terminal velocity of a 2 kg mass, dropped from a height of 220 m, with initial velocity -4 m/s, and experiencing a drag force proportional to the magnitude of its velocity, and having a constant of proportionality -50.
(2) It use to be legal to place toxic waste in sealed 55 gallon drums and dump them into the oceans for disposal. Theoretically, the drums were of such quality that they would not corrode, and their seals would not crack. However, given enough velocity, a drum could easily crack during its collision with the ocean floor.
Three forces act on a drum headed to the ocean floor: weight, drag, and buoyancy. Drag and buoyancy oppose a drum's progress to the bottom. Buoyancy has a magnitude equal to the weight of the water displaced. A 55 gallon drum of salt water weighs 2092 N. Assume that the drag force on a drum is linearly proportional to the drum's velocity, and that the constant of proportionality is known to be 1.17 (note that this is -1.17 if the positive direction of your vertical axis points up from the ocean floor).
Suppose each drum weighs 2346 N, and experimentation determines that it is likely to crack if it makes impact with a solid surface at velocities exceeding 20 m/s. If drums are disposed of in 350 m of water, are they likely to crack? (Assume the drums are designed against implosion at depths less than 500 m.)
Hint: Note that if the cracking velocity is greater than the terminal velocity, then their is no chance that a drum will crack upon impact. However, if this is not the case, then you need to determine if a drum will reach the bottom before if reaches it's cracking velocity. If the later is the case, you will find that although you can integrate to derive an equation for position as a function of time, you cannot use this equation to find an analytic inverse function that expresses time as an explicit function of position -- you have an equation that will tell you position for any time you plug into it, but you can't manipulate it into a form that will let you get at a vice versa state. Situations like this arise often and the techniques for dealing with them are time consuming. However, with a CAS like Maple these situations are not a problem, for Maple can find numeric solutions for these equations, easily providing the exact time at which the drum will be 350 m below sea level. This is quite a time saving feature of a tool like Maple.
Homogeneous Linear Diff Eqs Nonhomogeneous Linear Diff Eqs
A System with Air Resistance Top
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