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Unformatted text preview: — 1 — Problem. Path length (AEP) The simplest model for the flight of a scientific sounding rocket is that of a non-rotating, flat earth with no atmosphere. Using this model, we wish to determine the total path length of the trajectory for heights above 72 km. This is needed to determine the amount of chemiluminescent material (Tri-Methyl Aluminum – CH 3 Al, which burns in atomic oxygen) to deposit along the path so that the upper atmospheric winds can be determined by triangulation. An example of just such an experiment carried out by Cornell rocket scientists is shown below! For the sake of computational simplicity, we will take the acceleration due to Earth’s gravity to be constant at 10 m / s 2 . Newton’s Law, F = m a , can be split into two separate, second-order differential equations for the altitude z ( t ) and the ground range x ( t ) as follows: d 2 dt 2 z ( t ) =- g and d 2 dt 2 x ( t ) = 0 . In both cases the mass term has cancelled out. a) Indefinite integrals of the above equations will generate expressions for dz ( t ) /dt and dx ( t ) /dt involving one unknown constant each. Integrating once more will yield expressions for z ( t ) and x ( t ) involving one more constant each. Carry out these indefinite integrations below: Solution. d dt z ( t ) =- gt + C 1 d dt x ( t ) = C 3 z ( t ) =- gt 2 2 + C 1 t + C 2 x ( t ) = C 3 t + C 4 — 2 — To determine the four constants in the last two equations, we need to have two initial or “boundary” conditions for the system. These involve the position at t = 0 : [ x (0) , z (0)] = (0 , 0) and the velocity components at t = 0 : [ dx (0) /dt, dz (0) /dt ] = (400 m / s , 2000 m / s) ....
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- Fall '07
- Fundamental Theorem Of Calculus, m/s, path length