The Parachute Problem adapted from Douglas B Meades

The Parachute Problem (adapted from Douglas B. Meade’s “ODE Models for the Parachute Problem”) Noam Goldberg Craig Kaplan Tucker Riley

Outline l Introduction to the Parachute Problem l Modeling the Descent l Derivation of Equations l Application of Equations l Results

The Parachute Problem Newton’s Second Law: x x 0 v Figure 1: Forces acting on a skydiver in flight

Mathematical Model of Falling Object Or, in the case of the Parachute Problem:

Equivalent First-Order System

Solving for Velocity → ↓ (We can use an integrating factor!)

Integrating Factor

Finding the Integrating Constant (General velocity equation) Using the initial condition: We can solve for A:

Velocity Equation Before Deployment

Solving for Position Using our velocity equation, u-substitution, and the initial condition: we see that

Solving for Velocity After Deployment At t=td we have a new initial condition: Plugging this value into the general velocity equation (the one we had before plugging in ICs), we obtain: Solving for A, we get:

Solving for Velocity After Deployment l Plugging A into the original velocity equation, we find that: To get v 0, we plug t=td into our equation for velocity before deployment:

Solving for Velocity After Deployment ↓

Velocity Equation for Whole Jump

Applications of Derived Equations h = 3500 m td = 60 seconds 1. 2. 3. When the chute is pulled, what is the elevation and velocity of the skydiver? How long is the total jump? At what velocity does the skydiver hit the ground?

1. Elevation and velocity at time of deployment m m/s

2. Total length of jump Setting x(tf) = 0, solve for tf s

3. Final velocity Solve for v(116) m/s

Graph of negative velocity versus time -velocity (m/s) time (s)

Applications of Derived Equations For h = 3500 m td = 60 seconds x(td) = 325 m, v(td) = -58. 8 m/s ≈ 130 mi/hr 2. tf = 116 s 3. v(tf) = -5. 88 m/s 1.

Conclusion Verified that Meade’s equations are correct by deriving them ourselves l Used derived equations to find various velocities and positions, and total time of a typical jump (Meade) l FOR SALE: PARACHUTE ONLY USED ONCE NEVER OPENED SMALL STAIN

Works Cited l Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. Third Edition. Belmont, CA: Brooks/Cole, 2006. Print. l Meade, Douglas B. “ODE Models for the Parachute Problem. ” Siam Review 40. 2 (1998): 327 -332. Web. 27 Oct 2010.