(This project was contributed by and used with permission from Dr. Stephen Davies, University of Mary Washington (Davies 2012).) For a dangerous mission, a team of paratroopers must land on a 200-ft by 200-ft square roof of a 250-ft-tall building. The soldiers will be dropped at an altitude of 6400 ft from a transport jet traveling at a cruising speed of 200 knots. Certain restrictions apply to the jump: For safety, the paratroopers must not be airborne for more than 2 min and must hit the ground at less than 20 mi/h. Each paratrooper will be carrying a load of 40 lb of equipment, in addition to an assumed body weight of 180 lb. Before opening a chute, we can assume each soldier has a surface area of 0.4 m2 , in both horizontal and vertical directions. After opening, we can assume a 20-m2 surface area projected vertically, and a 1-m2 surface area projected horizontally. Assume good weather and negligible wind.

Develop a model for the jump. Plot altitude versus horizontal distance from the drop. Determine the following: The horizontal distance from the building the paratroopers should jump in order to land on the target area; the altitude at which they should pull their ripcords; the speed at which they will impact the rooftop; and the duration (in minutes and seconds) that they will be airborne. Use English units or metric units, but as with all models, be consistent throughout.

An important aspect of this problem that differs from the preceding skydiver model is that we must track and plot both x- and y-positions. In other words, in addition to following the troopers altitude over time, we must also monitor the horizontal distance from the drop point, so that we can plot the trajectory of the falling trooper. Time is still the key independent variable. However, we now need a stock (box variable) for the horizontal (x) position in addition to one for the vertical (y) position. Moreover, we should graph the y-position versus the x-position, or altitude versus the horizontal distance, not altitude versus time.

The following physics facts are also useful: When the troopers make the jump, their initial horizontal (x) velocity will be the same as that of the plane (200 knots). The troopers horizontal velocity can be considered separately from their vertical (y) velocity; in other words, neither one affects the other, so they can both be tracked independently. Air friction acts in the horizontal direction just as it does the vertical direction; that is, horizontal drag retards movement according to the formula, 0.65Av|v|, where v is the velocity in the horizontal direction.