Let us attempt to find the equation of motion of a rocket. Let be the fixed mass of the
rocket engine and the payload, and the total mass of the propellant contained
in the rocket's fuel tanks at time . Suppose that the rocket engine ejects the propellant
at some fixed velocity *relative to the rocket*. Let us examine the rocket at two
closely spaced instances in time. Suppose that at time the rocket and propellant, whose
total mass is , are traveling with instantaneous velocity . Suppose, further, that
between times and the rocket ejects a quantity of propellant of mass (*n.b.*,
is understood to be negative,
so this represents a positive mass) which travels with velocity (*i.e.*, velocity
in the instantaneous rest frame of the rocket). As a result of the fuel
ejection, the velocity of the rocket
at time is boosted to , and its total
mass becomes . See Fig. 51.

Now, there is zero external force acting on the system, since the rocket is assumed to be in outer space.
It follows that the total momentum of the system is a constant of the motion. Hence, we
can equate the momenta evaluated at times and :

(203) |

(204) |

(205) |

(206) |

(207) |

(208) |

(209) |

Let us now consider the factors which might influence the design of a rocket for use in interplanetary or
interstellar travel. Since the distances involved in such travel are vast, it is
important that the rocket's final velocity be made as large as possible, otherwise the journey
is going to take an unacceptably long time. However, as we have just seen, the factor which
essentially determines the final velocity of a rocket is the speed of ejection of
the propellant relative to the rocket. Broadly speaking, can never significantly
exceed . It follows that a rocket suitable for interplanetary or
interstellar travel should have as high an ejection speed as practically possible.
Now, ordinary chemical rockets (the kind which powered the Apollo moon program)
can develop enormous thrusts, but are limited to ejection velocities below
about
. Such rockets are ideal for lifting payloads out of the Earth's
gravitational field, but their relatively low ejection velocities render them unsuitable for
long distance space travel. A new type of rocket engine, called an *ion thruster*, is currently
under development: ion thrusters operate by accelerating ions electrostatically to great
velocities, and then ejecting them. Although ion thrusters only generate very small thrusts, compared to
chemical rockets, their much larger ejection velocities (up to 100 times those
of chemical rockets) makes them far more suitable for interplanetary or interstellar space travel.
The first spacecraft to employ an ion thruster was the
Deep Space 1 probe, which was launched from Cape Canaveral on October 24, 1998: this
probe successfully encountered the asteroid 9969 Braille in July, 1999.