(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

so i read morin's derivation of rocket equation propelled by photons now i want to try for relativistic mass ejection but i am having some problems

let subscript e denote quantities of ejected material and subscript of r denote quantities of rocket

2. Relevant equations

##

P = \gamma (P' + \frac{v E'}{c^2})\\

P = \gamma m v\\

E = \gamma m c^2

##

3. The attempt at a solution

syaing the rocket is currently moving at speed v then if ejects dm at exhaust speed u its mass becomes v + dv and mass of m-dm

then as viewed from the rocket is

conservation of energy becomes

##

E'_e = mc^2 -(m-dm)c^2 = dm c^2

##

as mass is not conserved other form of energy of the ejected mass maybe written as

##

E'_e = \gamma _ e m' c^2

##

where m' is the observed mass of ejected mass

so

##

dm = \gamma _e m'

##

thus the momentum observed from rocket is

##

p_e' = \gamma _e m' u = dm u

##

is that correct please correct me if its wrong

so as viewed from the ground frame is

##

P_e = \gamma (P'_e + \beta \frac{E'_e}{c})

##

now my question is if the gamma corresponds to speed v or v+dv similary for the beta

##

\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\\

\gamma = \frac{1}{\sqrt{1 - \frac{(v+dv)^2}{c^2}}}\\

##

which one is the correct one morin used the first one but why that

because by the time rocket measured the momentum of the ejected mass its already moving v + dv with respect to ground so shouldn't it be the second gamma

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# Homework Help: Deriving the relativistic rocket equation

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