AA311.Homework3.Solutions

AA311.Homework3.Solutions - AA311 Homework 3. Due October...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
AA311 Homework 3. Due October 17, 2011. Solution. Give all your answers in SI units. Problem 1. Recall that, when developing the equations for the ISA and the notion of “geopotential height,” we assumed that g ( h G ) g 0 . The geopotential height h was then defined such that ΔPE( h ) = ΔPE( h G ) h = ± r earth r earth + h G ² h G This is a “zeroth order” approximation of the true acceleration due to gravity g ( h G ) = g 0 ± 1 1 + ( h G /r earth ) ² 2 A “first order” approximation is g ( h G ) g 0 ± 1 - 2 h G r earth ² 1. Verify the correctness of this first order approximation by expanding f ( ± ) = ± 1 1 + ± ² 2 in a Taylor series about ± = 0. Show the first three terms (i.e., to quadratic order in ± ). 2. Develop an expression for a new altitude ˜ h ( h G ) defined such that ΔPE( ˜ h ) = ΔPE( h G ) where ΔPE( ˜ h ) = Z ˜ h 0 m g 0 1 - 2 ˜ h r earth ! d ˜ h Solution. Recall that the Taylor series expansion of a function f ( ± ) about ± = 0 is f ( ± ) = X n =0 1 n ! d n f n ³ ³ ³ ³ ± =0 ± n = f (0) + df ³ ³ ³ ³ ± =0 ± + 1 2 d 2 f 2 ³ ³ ³ ³ ± =0 ± 2 + ... Thus, f ( ± ) = 1 + ± - 2 ± 1 1 + ± ²² ± =0 ± + 1 2 ± 2 ± 1 1 + ± ²² ± =0 ± 2 + ... = 1 - 2 ± + ± 2 + ... Integrating the potential energy function given for ˜ h gives ΔPE( ˜ h ) = Z ˜ h 0 m g 0 1 - 2 ˜ h r earth ! d ˜ h = m g 0 ˜ h 1 - ˜ h r earth ! Setting this equal to ΔPE( h G ) = m g 0 h G ± r earth r earth + h G ² 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
0 200 400 600 800 1000 6.5 7 7.5 8 8.5 9 9.5 10 Geometric Altitude (km) Acceleration Due To Gravity (m/s 2 ) True Constant Model
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

AA311.Homework3.Solutions - AA311 Homework 3. Due October...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online