Math 216 Demonstrations

Skydiver Velocity Model, Numerical Methods

Abstract

We continue analysis of the skydiver model of the 2_3Skydiver demonstration, applying the improved Euler and Runge Kutta methods. Again, the discontinuity in the resistance term provides for a discernable deviation between the numerical methods and between their approximations and the exact solution.

Use Cases

Lecture: It is probably instructive to work a few steps of the improved Euler method to see how it compares with the Euler calculations. It may or may not be useful to similarly work steps of RK. The demonstrations show the difference in the numerical methods, and in their errors.

Outside of Lecture Find the first four approximations to the solution to the initial value problem with the improved Euler's method and \(h=0.6\). Check that your solution matches the values shown in the demonstration, and that you understand the errors illustrated there. Work out the first three steps of the approximation to the solution to the initial value problem using the Runge Kutta method and \(h=0.6\).

Model Description

We again model the velocity of a skydiver by considering her/his motion in two phases, as indicated in 2_3Skydiver, giving the model \[ \frac{dv}{dt} = -g + F_R, \] where \(g\) is the acceleration due to gravity and \(F_R\) is the piecewise defined force of air resistance divided by the mass of the skydiver. We take \[ F_R = \begin{cases} k_1\,v^2 & \mbox{before deployment}\\ -k_2\,v & \mbox{after deployment}. \end{cases} \]

As before, we note that here we take upwards to be the positive direction, so that the force of gravity pulls in the negative direction and \(F_R > 0\) opposes that motion.

ODE Model

Using data from [1], we have (as described in 2_3Skydiver) \[ \frac{dv}{dt} = -32.2 + \begin{cases} 0.00104\,v^2 & \mbox{before deployment}\\ -2.01\,v & \mbox{after deployment} \end{cases}. \] We take \(v(0) = 0\).

Euler's, the improved Euler and Runge Kutta methods are applied in a straightforward manner.

Matlab Demos

We consider a number of Matlab demos for this. These require the file arrow.m (downloads as a zip file with the Matlab file and license) to be in the same directory as the demonstration file, or in the Matlab path.

Looking at the Model

Some questions that may be worth considering:

References

  1. Meade, D.B. ODE Models for the Parachute Problem. SIAM Rev. 40(2), 327-332 (1998).
UMMath Math 216 Lecture Demos: 2_5NumMethods
Last Modified: 03:00 EST 01/02/2013
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