lecture3b

# lecture3b - Rocket Equation The rocket equation How much...

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Rocket Equation Konstantin Tsiolkovsky The rocket equation: How much fuel is required? Rocket equation , derived from conservation of momentum, trades off exhaust velocity with payload fraction. It was first derived by Tsiolkowsky in 1895 for straight-line rocket motion with constant exhaust velocity. It is also extended for elliptical trajectories and applied to chemical and nuclear-thermal rockets. Propellant gets used up as the vehicle produces thrust to accelerate. Also, as the propellant is blasted out the exhaust nozzle, the mass of the vehicle decreases.

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Rocket Equation Konstantin Tsiolkovsky The rocket equation: How much fuel is required? The ideal rocket equation estimates the amount of propellant needed to change a vehicle’s velocity Δ v. It relates mission requirements to system requirements .
MV = Δ M[V-U ex ] + [M- Δ M] [V+ Δ V] MV = Δ MV- Δ MU ex +MV - Δ MV+M Δ V- Δ M Δ V M Δ V= Δ MU ex + Δ M Δ V dt V M d F external ] [ r = Newton’s Third Law Look at a rocket in horizontal flight

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F = Engine mass flow Engine thrust equation We shrink time as small as possible Reaction Force on Rocket Newton’s Third Law Dividing by Δ t and evaluating limit { Δ M, Δ V, Δ t} 0 ex ex U dt dM dt dV M t V M U t M t V M = Δ Δ Δ + Δ Δ = Δ Δ as limit { Δ M, Δ V, Δ t} 0 ex U dt dM dt dV M = = m propellant C e
Compare the results F= Using Fluid Conservation laws Using Newton’s laws Effective Exhaust Velocity F = m e V e + p e A e p A e () U ex = V e + p e A e p A e m e C e ex U dt dM dt dV M =

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lecture3b - Rocket Equation The rocket equation How much...

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