Relativistic Rocket

A rocket having initially a total mass M₀ ejects its fuel with constant velocity –u (u> 0) relative to its instantaneous rest frame. According to  Newtonian mechanics, its velocity V , relative to the inertial frame in which it was originally at rest, is related to its mass M(V) by the formula

a) Derive this result.

b) Suppose the velocity of the ejecta is limited only by 0 ≤ u ≤ c, and derive the relativistic analogue of the above equation. Show that it reduces to the Newtonian result at the appropriate limit.

SOLUTION

a) Let us consider the short interval dt in the center of mass frame moving with velocity v; the fuel is ejected with velocity u in this frame. At time t' = t + dt the velocity of the rocket increases by dv. The mass M(t) of the rocket decreases by dM(dM < 0) and the mass |dM| of the ejected fuel will have a velocity -u in this frame. Momentum conservation gives

(1)

where M(t + dt) is the mass of the rocket at time t + dt. Expanding M(t + dt) as

and neglecting second-order terms in the differentials yields

(2)

Transforming to the lab frame and using dv = dV, where V is the velocity of the rocket in the lab frame, we obtain a solution for the initial condition v(0) = 0:

(3)

b) Write down momentum conservation in the rocket’s frame:

(4)

where M is the mass of the rocket, dm is the mass of the fuel, and   Energy conservation in the frame of the rocket gives

(5)

We ignored the relativistic corrections to the mass of the rocket in (5) and terms such as  in (4). Substituting from (5) into (4), we have

(6)

which is the same result as obtained in the nonrelativistic calculation of (a). Now we must transform dv from the instantaneous rocket frame to the laboratory frame. Using the equation for the addition of velocities, we have

(7)

where V + dV is the new velocity of the rocket in the lab frame. Rearranging (7) gives

(8)

where  Substituting (8) into (6), we obtain Mγ² dV = -u dM or

(9)

Integrating (9), we have

(10)

where β = V/c, from which we find

(11)

If β << 1 then (11) boils down to

the same result as that obtained in (a).