2 D. B. MEADE

Textbook  k_{1}/m k_{2}/m  t_{d}  x_{0}
Abell/Braselton  1/6  5/3  60sec  n/a
Drucker  4/15  4/3  x(14) = 0  1200ft
Edwards/Penney  3/20  3/2  20sec  10,000ft
Kostelich/Armbruster 1/5 ≈ 1.56 v(t_{d}) = 0.95v_{∞} 2000m
Nagle/Saff  1/5  7/5  60sec  4000m

Table 2.1

Coefficients of air resistance, deployment criteria, and jump height for the parachute problem.

improved model. Additional extensions of the problem are included in the concluding remarks in Section 4.

2. The Basic Model. The motion of the skydiver is governed by Newton’s Second Law of Motion. Balancing the forces of acceleration, gravity, and air resistance yields the second-order (linear) initial value problem

mx′′ = −mg − kx′,  x(0) = x_{0},  x′(0) = 0

where x is the height of the skydiver above the Earth’s surface, m is the skydiver’s mass, g is the gravitational constant, k is the coefficient of air resistance, and prime (′) denotes differentiation with respect to time. An equivalent first-order system for the position and velocity,^{1} v, is

\begin{aligned}v′ & = & −g − \frac{k}{m}v, & v(0) = 0, \\ x′ & = & v, & x(0) = x_{0}. \end{aligned} (2.1)

This IVP has the advantage that the velocity equation (2.1)_{1} can be solved as a firstorder linear ODE; the position is then obtained by integration. Almost any ODE text contains the explicit solution for the case when m, g, and k are constants:

v(t) = \frac{mg}{k} (e^{− \frac{k}{m} t} − 1) , x(t) = x_{0} + \frac{m^{2}g}{k^{2}} (−\frac{k}{m}t + (1 − e^{− \frac{k}{m} t})) . (2.2)

But, k is not constant in the parachute problem. The statement of the problem suggests the general form for the coefficient of air resistance is:

k(t) = \{ \begin{matrix}k_{1}, & t < t_{d} \\ k_{2}, & t ≥ t_{d} \end{matrix} (2.3)

where t_{d} is the time when the parachute is deployed. The form of the differential equation suggests that k/m, with units 1/time, can be considered in place of the two parameters k and m. Note that this observation eliminates many of the potential problems that arise from the mixing of the CGS and MKS systems. Typical parameter values found in several ODE textbooks are reported in Table 2.1.

Note that if t_{d} is a function of velocity, e.g., deployment occurs when the velocity reaches a specified threshold, or position, e.g., deployment occurs at a given altitude, then the IVP is nonlinear. Typically, it will be necessary to find the solution with k = k_{1}, compute t_{d}, then solve the problem with k = k_{2} and initial conditions selected to enforce continuity of position and velocity at the time of deployment: v(t_{d}^{+}) = v(t_{d}^{−})

1Note that v < 0 for a body falling towards Earth.