PARACHUTE PROBLEM 5

(a) 0 ≤ t ≤ 200 seconds  (b) 9 ≤ t ≤ 15 seconds

Fig. 3.1. Motion for (a) 0 ≤ t ≤ 200 seconds and (b) 9 ≤ t ≤ 15 seconds for a C^{0} coefficient of air resistance.

(a) 0 ≤ t ≤ 200 seconds  (a) 9 ≤ t ≤ 15 seconds

Fig. 3.2. Motion for (a) 0 ≤ t ≤ 200 seconds and (b) 9 ≤ t ≤ 15 seconds for a smooth (C^{1}) coefficient of air resistance.

Assuming t_{d} and τ_{d} are known, or can be determined, each problem is a first-order linear ODE. In this sense the modified model is no more complicated that the original.

To complete the model it is necessary to choose a specific k_{d}. Continuity of the acceleration will be assured when k_{d} interpolates the two constant states. For example, when k_{d} is a linear function, the motion (position, velocity, acceleration, and jerk) appear as in Figure 3.1. Note that this model predicts the maximum jerk is under the 3 G threshold, but occurs at the instant the ripcord is pulled. To obtain a smooth jerk k_{d} must be selected so that the derivatives at t = t_{d} and t = t_{d} + τ_{d} both vanish. A cubic function can easily be fit to these conditions; the resulting motion is displayed in Figure 3.2. Note that this motion is consistent with all of the characteristics found in the Air Force Academy Training Manual [8]: snatch force is smooth and well below the 3 G threshold; deployment begins at an altitude of 928 m (3045′ ) AGL; an additional 3 seconds of free-fall brings the altitude to ≈2500′ — thus the strict penalty for free-falls longer than 13 seconds. The complete jump lasts a little more than three minutes (196 seconds) with a landing velocity of 4.6 m/s (15.1 ft/sec).