Symbolic Math Toolbox

Modeling the Velocity of a Paratrooper

Introduction

We will model the velocity of a paratrooper that is dropped from an airplane. 

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The forces acting on the paratrooper include gravitational force and the opposing drag from the parachute.  The governing equation can be expressed as follows.

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where:

m - paratrooper mass (kg)

g - acceleration due to gravity (m/s²)

V(t) - paratrooper velocity (m/s)

K1 - drag constant, assumed to be 40

Define and Solve System Equation

We define the paratrooper equation and solve it analytically.

eq:= m*diff(V(t),t) + m*g - 40*V(t)^2

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velocity := solve(ode({eq,V(0) = 0},V(t)),IgnoreAnalyticConstraints)

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Since acceleration due to gravity and paratrooper mass are both greater than zero, we extract the second element of our piecewise solution.

Result:= Simplify(velocity) assuming g > 0 and m > 0

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velocity:= Result[1]

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Plot Paratrooper Velocity

The paratrooper mass is 80 kg, and acceleration due to gravity is 9.81 m/s2.  We plug these values into the velocity equation and plot the result.

plot(velocity | m=80 | g=9.81 , t = 0..10)

MuPAD graphics

We see that the paratrooper velocity begins to reach steady state at around 1.5 seconds.  This is when the drag force from the parachute is roughly equivalent to the gravitational force, and there is no further acceleration. 

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