Question

A relativistic rocket emits a gas jet with non-relativistic velocity u constant relative to the rocket. Find how the velocity v of the rocket depends on its rest mass m if the initial rest mass of the rocket equals to `m_0`.

Answer 1

The velocity of ejected gases is u relative to the rocket. In an earth centred frame it is
`(v-u)/(1-(vu)/(c^2))`
in the direction of the rocket. The momentum conservation equation then reads
`(m+dm)(v+dv)+(v-u)/(1-(uv)/(c^2))(-dm)=mv`
or `mdv-((v-u)/(1-(uv)/(c^2))-v)dm=0`
Here `-dm` is the mass of the ejected gases. so
`mdv-(-u+(uv^2)/(c^2))/(1-(uv)/(c^2))dm=0`, or `mdv+u(1-v^2/c^2)dm=0`
(neglecting `1-(uv)/(c^2)` since u is non-relativistic.)
Integrating `(beta=v/c), int (dbeta)/(1-beta^2)+u/cint(dm)/(m)=0, 1n(1+beta)/(1-beta)+u/c1nm=const ant`
The constant `=u/c1nm_0` since `beta=0` initially.
Thus `(1-beta)/(1+beta)=(m/m_0)^(u//c)` or `beta=(1-(m/m_0)^(u//c))/(1+(m/m_0)^(u//c))`