ISSUES IN SOLUTION

Many students may (correctly) use the same time variable through out rather than starting each IVP at time zero as presented here.

In (3), the solution is sensitive to the transition time period lengths. The good students should determine just how sensitive it is in reasonable time ranges.

It is far more important that the students are allowed time to grapple with modeling issues instead of the method of solution. While the methods involved with solving intermediate parts of the problem are interesting in their own right, if the students are forced to dwell on the details in this problem, they will never get to the major issues of the problem. Thus, use of a CAS such as Mathematica or Maple to solve the DEs quickly is advisable.

Current computer algebra systems (CASs) can actually make modifications of this problem harder for students who use them! A typical extension (see Extensions) is to assume that air resistance is proportional to the square of velocity so that the model for each phase of flight is
m dv/dt = mg - k v^2.
Given a CAS, students will be inclined to solve as before, but by doing so blindly, the complete solution cannot be obtained due to the fact that the CAS doesn't solve the ODE over the appropriate regions for each phase of flight unless sufficiently coaxed. (Note that obtaining the solution of the ODE entails integrating
m/(mg - k v^2)
with respect to v.) The solution can be represented implicitly by logarithms or by hyperbolic tangents. Either way, the relationship between the initial conditions for each phase of flight and the drag coefficient k affects the form of the solution.
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