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Relativistic Rocket, Its Equation of Motion and Solution for two Special Cases
Somnath Datta
This article adopts 4-dimensional Minkowski formalism to obtain the equation of motion of a relativistic rocket, i.e., a vehicle in which the exhaust particles are ejected with a fixed relativistic velocity $u$ opposite the direction of the motion of the rocket, which is taken to be the $x$ direction. We obtained the equation of motion in the instantaneous rest frame of the rocket, labeled as $Sz$, and then converted this equation of motion into the ground frame $S$. As a prerequisite to this derivation we also reviewed the mass equation of the rocket, i.e., the relationship between the instantaneous rest mass $M$ of the rocket and its velocity $v$. We subjected both equations, i.e., the mass equation and the equation of motion to the N.R. test, i.e., the requirement that the forms they assume when $v ll c$ (where $c$ is the velocity of light) converge to their Non Relativistic counterparts. We obtained the solution of the equation of motion in two special cases, namely (i) $u=c/3$, and (ii) $u=c$, and made a plot of the $v-t$ relationship for both cases. It is seen that the $v-t$ plot for the case (i) nearly follows the corresponding N.R. counterpart, i.e., $u ll c$, up to $v simeq 0.5 c$.
Special Relativity, Minkowskian Equation of Motion