Classical Mechanics (Goldstein) · Problem 23

The Classical Mechanics of the Special Theory of Relativity — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · relativistic-rocket-motion

Problem

The theory of rocket motion developed in Exercise 13, Chapter 1, no longer applies in the relativistic region, in part because there is no longer conservation of mass. Instead, all the conservation laws are combined into the conservation of the world momentum; the change in each component of the rocket's world momentum in an infinitesimal time

dt

must be matched by the value of the same component of

p_v

for the gases ejected by the rocket in that time interval. Show that if there are no external forces acting on the rocket, the differential equation for its velocity as a function of the mass is

m\frac{dv}{dm} + a\left(1 - \frac{v^2}{c^2}\right) = 0,

where

a

is the constant velocity of the exhaust gases relative to the rocket. Verify that the solution can be put in the form

\beta = \frac{1 - \left(\frac{m}{m_0}\right)^{2a/c}}{1 + \left(\frac{m}{m_0}\right)^{2a/c}},

m_0

being the initial mass of the rocket. Since mass is not conserved, what happens to the mass that is lost?

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