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MathematicalPhysicist
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problem statement
citation: introduction to SR by wolfgang rindler, second edition, page 96-97 question number 6.
A rocket propels itself rectilinearly by giving portions of its mass a constant backward velocity U relative to its instantaneous rest frame.
It continues to do so until it attains a veclocity V relative to its initial rest frame.
prove that the ration of initial to the final rest mass of the rocket is given by:
[tex]\frac{M_i}{M_f}=(\frac{c+V}{c-V})^{\frac{c}{2U}}[/tex]
attempt at solution
well I am given the hint that:
[tex](-dM)U=Mdu'[/tex]
where: M is the rest mass of the rocket and u' is its velocity realtive to its instantaneous rest frame.
well i think that the equation for the mass is: [tex]M(t)=M_i-dM\frac{dt}{d\tau}[/tex]
where: [tex]\frac{dt}{d\tau}=\gamma(u')[/tex]
so the equation (erroneously) transforms to:
[tex]-UdM\gamma(u')=Mdu'[/tex] adn then one only needs to integrate between the masses: M_i upto M_f and from 0 to V in the velocities side of the equation, but i don't get the required answer, any pointers here on where i got it wrong?
thanks in advance.
citation: introduction to SR by wolfgang rindler, second edition, page 96-97 question number 6.
A rocket propels itself rectilinearly by giving portions of its mass a constant backward velocity U relative to its instantaneous rest frame.
It continues to do so until it attains a veclocity V relative to its initial rest frame.
prove that the ration of initial to the final rest mass of the rocket is given by:
[tex]\frac{M_i}{M_f}=(\frac{c+V}{c-V})^{\frac{c}{2U}}[/tex]
attempt at solution
well I am given the hint that:
[tex](-dM)U=Mdu'[/tex]
where: M is the rest mass of the rocket and u' is its velocity realtive to its instantaneous rest frame.
well i think that the equation for the mass is: [tex]M(t)=M_i-dM\frac{dt}{d\tau}[/tex]
where: [tex]\frac{dt}{d\tau}=\gamma(u')[/tex]
so the equation (erroneously) transforms to:
[tex]-UdM\gamma(u')=Mdu'[/tex] adn then one only needs to integrate between the masses: M_i upto M_f and from 0 to V in the velocities side of the equation, but i don't get the required answer, any pointers here on where i got it wrong?
thanks in advance.