Question
A rocket is initially stationary with a rest mass $M_i$ in the lab frame $\mathrm{S}$. The rocket then begins to convert mass into photons and ejects them from the back. When the rest mass of the rocket is $M_f$, prove that its speed $u$ in frame S fulfils$$\frac{M_i}{M_f}=\left(\frac{1+u}{1-u}\right)^{\frac{1}{2}} .$$
Step 1
Step 1: Conserve four-momentum: in lab frame S, initial four-momentum is (E_i/c, p_i) = (M_i c, 0). Show more…
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Key Concepts
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Photon rocket. A "photon rocket" uses pure radiation as the propellant. If the initial and final rest masses of the rocket are Mi and Mf, show that the final speed v of the rocket relative to its initial rest system is given by the equation, Mi/Mf = sqrt((c + v)/(c - v)).
(a) A rocket of (variable) mass $m$ is propelled by steadily ejecting part of its mass at velocity $u$ (constant with respect to the rocket). Neglecting gravity, the differential equation of the rocket is $m(d v / d m)=-u$ as long as $v \ll c, c=$ speed of light. Find $v$ as a function of $m$ if $m=m_{0}$ when $v=0$ (b) In the relativistic region (v/c not negligible), the rocket equation is $$ m \frac{d v}{d m}=-u\left(1-\frac{v^{2}}{c^{2}}\right) $$ Solve this differential equation to find $v$ as a function of $m$. Show that $v / c=(1-x) /(1+x)$, where $x=\left(m / m_{0}\right)^{2 u \mid \text {. }}$
ORDINARY DIFFERENTIAL EQUATIONS
Miscellaneous problems
Suppose that at time $t$, the rocket has the mass $m$ and the velocity $\vec{v}$, relative to the reference frame, employed. Now consider the inertial frame moving with the velocity that the rocket has at the given moment. In this reference frame, the momentum increament that the rocket \& ejected gas system acquires during time $d t$ is, or, $$ \begin{gathered} d \vec{p}=\dot{m} d \vec{v}+\mu d t \vec{u}=\vec{F} d t \\ m \frac{d \vec{v}}{d t}=\vec{F}-\mu \vec{u} \end{gathered} $$ or, $$ m \vec{w}=\vec{F}-\mu \vec{u} $$
Physical Fundamentals Of Mdchanics
Laws of Conservation of Energy, Momemtum, and Angular Momentum
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