The Parachute Problem

A man with a parachute jumps out of an airplane at an altitude of 5000 feet.  After 5 seconds his parachute opens.  During freefall the man's drag coefficient is 1 and air resistance is taken as proportional to velocity, i.e., R = kv = 1v.  After the parachute opens the drag coefficient is 5.  The total weight of the man and his parachute is 200 pounds.  Take the initial vertical velocity when he jumped out of the plane to be zero.  Approximately how far did the man fall during his 5 seconds of freefall?  What was his vertical velocity the instant his parachute opened?  Approximately what was the total amount of time it took him to reach the ground from the instant he jumped out of the airplane and what was his vertical velocity when he reached the ground?  The instant the first man's parachute opened a second man jumped out of the airplane at the same altitude.  The second man has the same weight and drag coefficient as the first man.  Regrettably the second man's parachute and back-up parachute both malfunctioned and did not open.  Approximately how long before the second man reaches the ground and what is his vertical velocity when he reaches the ground?  What is the limiting velocity for each man (for the first man after his parachute opens)?

The limiting velocity is W/k.  Enlargement

Integrate in (3) to find the position function s(t).  Down is positive in this model.  Initially you will need to find the velocity and position functions for freefall.  (Take v(0) = 0 and s(0) = 0.)  You will then need to find the velocity and position after 5 seconds (at t = 5).  Assume the parachute opens instantly.  You will then need to find the velocity and position functions for the parachutist using the initial velocity and initial position given by what you found to be velocity and position after 5 seconds of freefall.  Once this is done you can approximate how long it will take for him to reach the ground and what his velocity will be when he reaches the ground.  Take g = 32.

Click here for a (simple) animation of the descent of the parachutist whose parachute does open.  I will need to model another example where the length of time of the freefall is longer for a better effect in the animation.  Here is most of the correct solution.  You can determine the limiting velocities.  There are errors in the solution manual for section 3.1:  32, a problem similar to this one.  In the solution manual the constant of integration in going from the velocity function to the position function has been ignored, leading to errors.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats