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We want to model the path a skydiver will
take after jumping out of a plane.
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To do this we give the system some
parameters, such as the wind speed in the x,y
and z directions, gravity, mass of the
skydiver, drag coefficient, temperature and
density of the air.
We treat the skydiver as a spherical object for
simplicity.
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We want to get the path the skydiver will
take. We will need position, velocity and
acceleration vectors in the x,y and z
directions over small time intervals.
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The acceleration of the skydiver is given by;
where
here, ρ is air density, A is the cross sectional area of the skydiver, Cd is
the drag coefficient. The velocities are taken relative to the component
wind velocities, they equal the velocities of the skydiver minus the
velocities of the wind.
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We want to reduce that system of 2nd order
ODE’s into a system of 1st order ODE’s. So we
introduce a state variable y such that;
Where x,y,z are the positions, and
Vx,Vy,Vz are the velocities of the
Skydiver.
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We then differentiate the vector y to get;
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We now need some values for the parameters
and then we can solve the equation.
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We also need values for the parameters in y
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Using the parameters for dy/dt we set up a
code to solve it named myskydiver.m
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Using myskydiver.m, the parameters for y
and the built in ode5 solver we can obtain the
position vectors for the skydiver. Here is the
code for runskydive.m
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Between dy/dt and y, we have the position,
velocity and acceleration components. We can
plot many different aspects of the skydiver’s
path.
Of interest might be the z velocity component
vs time, or a 3D plot of the x,y and z
components.
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You can alter several things in this model
such as varying wind speed with altitude and
the drag constant.
By altering the drag constant, we can
simulate the moment when the skydiver pulls
his parachute.
We then run the codes from earlier for the
time when he pulls the parachute until when
he hits the ground.