1. Skydiver

2. The Problem
We want to model the path a skydiver will take after jumping out of a plane.

3. The Model
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.

4. What do we want?
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.

5. The Set up 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.

6. The Set up
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.

7. The Set up
We then differentiate the vector y to get;

8. The Set up
We now need some values for the parameters and then we can solve the equation.

9. The Set Up
We also need values for the parameters in y

10. The Code
Using the parameters for dy/dt we set up a code to solve it named myskydiver.m

12. The Code
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

13. The Code
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.

14. 3D plot of x,y and z

15. Using the Parachute
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.

16. 3D plot of x,y,z with parachute