Maple and the Parachute Problem: Modelling with an Impact

Douglas B. Meade^{∗}

Abstract

The “parachute problem” involves a first-order ODE in which the coefficient of air resistance is a piecewisedefined function. This paper focuses on the use of Maple’s symbolic, numeric, and graphic tools to facilitate the analysis of this initial value problem. Of particular interest is a qualitative analysis of the solution. The form of the problem most commonly found in textbooks is shown to be physically unrealistic. An improved model is proposed and analyzed. The paper concludes with the analysis of parameter sensitivity of the model and the solution of an interesting control problem. The graphical analysis is based on two Maple procedures, motion and plot_motion, which are defined in this paper.

Introduction

The “parachute problem” is based on a model for the motion of a skydiver when the coefficient of air resistance changes between free-fall and final descent. The author has previously shown that many of the problems posed in textbooks are not physically realistic and has proposed an improved — but not more complicated — based on real-life information about skydiving[6]. While the basic problems are not too sophisticated, manual computation of the solution is prohibitive. As a result, much of the analysis will be based on information obtained directly from the differential equation. The purpose of this paper is to illustrate some of the ways in which Maple can be used to facilitate the analysis; see

[6] for a more mathematical perspective of the problem.

The Original Model

The “parachute problem” will be considered as a system of two initial value problems for the position, x, and (vertical) velocity, v, of a skydiver under the forces of gravity and air resistance. When the force due to air resistance is proportional to the velocity of the parachutist the model is:

\begin{aligned}v′ & = & −g − \frac{k}{m}v, & v(0) = 0, \\ x′ & = & v, & x(0) = x_{0} \end{aligned}  (1)

Reference  k_{1}/m k_{2}/m  t_{d}  x_{0}
[1, p. 141, #20]  1/6  5/3  60sec  n/a
[3]  4/15  4/3  x(14) = 0 1200ft
[4, p. 95, #10]  3/20  3/2  20sec  10,000ft
[5, p. 109, #20]  1/5 ≈ 1.56 v(t_{d}) = v^{∗} 2000m
[9, p. 112, Ex. 3] 1/5  7/5  60sec  4000m

Table 1: Coefficients of air resistance, deployment criteria, and jump height found in different versions of PP.

where g is the gravitational constant, k is the coefficient of air resistance, m is the mass of the skydiver, and x_{0} is the altitude at which the jump begins. The coefficient of air resistance is frequently modelled as a piecewise constant function ([1], [3], [4], [5], [9]):

k(t) = \{ \begin{matrix}k_{1}, & t < t_{d} \\ k_{2}, & t ≥ t_{d} \end{matrix}  (2)

where t_{d} is the time when the parachute is deployed.^{1} Parameter values found in a variety of textbooks are summarized in Table 1.

To prepare for the Maple analysis, the plots package is needed:

                                                         > restart; with( plots ):

The basic problem to be studied is

                                                         > SYS := { diff( v(t), t ) = -g - k/m*v(t),
                                                         >  diff( x(t), t ) = v(t) };

SYS := \{\frac{∂}{∂t} v( t) = −g − \frac{k v( t)}{m}, \frac{∂}{∂t} x( t) = v( t)\}

with initial conditions

                                                         > IC := { v(0) = 0, x(0) = x0 };

IC := { x( 0 ) = x0 , v( 0 ) = 0 }

As the system of ODEs and initial conditions are both sets, it will be useful to have a means to extract a specific entry from either object.

                                                         > ic := (IVP,y0) -> op( select( has, IVP, y0 ) ):
                                                         > ode := (IVP,y) ->
                                                         >  op( select( has, IVP, diff(y(t),t) ) ):

The position (height above ground level), velocity, acceleration, and jerk (rate of change of acceleration)

\overline{1}

∗Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA; E-mail: meade@math.sc.edu; WWW URL: http://www.math.sc.edu/˜ meade/

1Note that t_{d} may be specified implicitly, e.g., deployment occurs when the velocity is 95% of the terminal free-fall velocity [5], in which case the IVP may not be linear.