Two Equation of Rocket

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Start from:

Pi=PfP_i = P_f

That :

{Mv=−dM⋅U+(M+dM)(v+dv)v+dv=vrel+U⇒−Mdv=dMvrel\begin{cases} Mv = -dM \cdot U + (M+dM)(v+dv) \\ v+dv = v_{rel} + U \end{cases}\\ \Rightarrow -Mdv = dM v_{rel}

First rocket equation

Differentiate :

dMdtvrel=−M⋅dvdt⇒T=Rvrel=M⋅a\frac{dM}{dt} v_{rel} = -M \cdot \frac{dv}{dt}\\ \Rightarrow T = R v_{rel} = M \cdot a

(RR is the rate of mass consumption)

This is the first rocket equation

(used in instantaneous moment).

Second rocket equation

Integrate :

∫vivfdv=−vrel∫MiMfdMM⇒vf−vi=vrelln⁡MiMf\int_{v_i}^{v_f} dv = - v_{rel} \int_{M_i}^{M_f} \frac{dM}{M}\\ \Rightarrow v_f - v_i = v_{rel} \ln \frac{M_i}{M_f}

This is the second rocket equation
(used in process).