Chap3_9_17 - AAE 439 Ch3 –9 IDEAL ROCKET EQUATION Assumptions Propellant flow rate is constant Gravity-free No drag Newton’s 2 nd Law

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Unformatted text preview: AAE 439 Ch3 –9 IDEAL ROCKET EQUATION Assumptions: Propellant flow rate is constant, Gravity-free, No drag. Newton’s 2 nd Law: Instantaneous Vehicle Mass: Velocity Change: Velocity History: Velocity Gain due to Burn of Propellant Mass M P during Burn Time t b : dm M u u e p e A e F ∑ = M du dt du = F M dt = u eq m M dt M t ( ) = m − m t = m 1 − t t b m P m ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ du = u eq α t b ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 − α t t b ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ dt Δ v = − u eq ln 1 − α ( ) = u eq ln m m f ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ e Δ v u eq = m m f = ℜ CG dm = + m dt = − dM dt dt du = − u eq dM M v ( t ) = du v i v f ∫ = − u eq dM M m m ∫ = u eq α t b 1 − α t t b ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 1 t ∫ dt AAE 439 Ch3 –10 IDEAL ROCKET EQUATION Ideal Rocket Equation: Usually, mission requirements are stated in terms of velocity increment, or ∆ v (“delta V”). The rocket equation defines the velocity increment, and tells us how much propellant is needed to change the velocity of a mass while its mass is changing (rockets naturally change mass as they expel burnt propellant). Examples: The ∆ v needed to put an object into LEO is about 36,000 ft/s ( 11 km/s ). For space missions, the achievable ∆ v tells you how long it will take to get to a far away place, and how much propellant you have to pack away to get there....
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Chap3_9_17 - AAE 439 Ch3 –9 IDEAL ROCKET EQUATION Assumptions Propellant flow rate is constant Gravity-free No drag Newton’s 2 nd Law

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