On-target home ==> Math modeling ==> Sizing a Parachute

by Levi Westra

## Goal

Determine the size of a parachute needed to deliver a specified payload at a specified speed

## Basic Concepts

When released from rest, a parachute will accelerate before reaching a steady speed.  This steady speed is called terminal velocity.  Terminal velocity is a consequence of equilibrium. That is, the sum of the forces on the parachute is equal to zero.  Forces acting on a parachute are shown in Fig. 1.

Figure 1

Weight (W) acts downward, through the center of mass.  The resistance of the ambient air creates a drag force D.  These forces are balanced once terminal velocity is reached.

W = D

## Weight

Weight is given by W = mg, where m is the mass of the payload plus the mass of the parachute.

## Drag

Drag force is the retarding force acting upon a body as it moves through a fluid.  Engineers commonly predict drag using.

where Cd, the coefficient of drag, is found from experiments.  Values for a parachute are 0.8 < Cd < 1.2.  Note that Cd is unitless.

The term Ap is the projected area of the object.  Imagine shining a light directly on an object.  The surface area of the shadow equals projected area.  Thus, for a hemisphere, the projected area equals the area of circle of radius r.

## Calculation of radius for a Parachute

To find the radius of a parachute, we equate weight with drag:

Using the drag formula gives