Question

A rocket is intended to leave the Earths Gravitational Field.The Fuel in its Main engine is a little less than the amount that is necessary, and an auxiliary engine, only capable of operating for a short time, has to be used as well.When is it best to switch on the Auxiliary engine : at take off, or when the rocket has nearly stopped with respect to the Earth, or does it not matter? Give your reasons.

Answer 1

Let's consider two things.

First, that the main and auxiliary rockets produce a set amount of impulse.
Second, that by burning the auxiliary rocket, we reduce the mass of the vehicle.

Let's define force and impulse as\[F = {dp \over dt} = {d (m \cdot v) \over dt}\]\[I = \int\limits F dt = \int\limits {dp \over dt} dt = \int\limits dp\]\[\int\limits F dt = \int\limits d(m \cdot v)\]

We need to make a couple of assumptions. First, let's say that the main rocket burns fuel at a greater rate than the auxiliary rocket. Second, let's say that the main rocket produces more thrust force than the auxiliary rocket.

If we fire the auxiliary rocket at launch, the main engine won't have to carry that additional mass of the auxiliary rocket up into orbit. Therefore, the main engine will be able to create a greater \(\Delta v\) on the vehicle in this case than if it had to carry the auxiliary rocket into orbit.

However, if we fire the auxiliary rocket after the main engines burn, the auxiliary rocket will create a greater \(\Delta v\) in this case.

I would recommend firing the auxiliary rocket first, since the main engine produces the greatest thrust and longest burn time.

Consider that the Space Shuttle fires its auxiliary rockets at the time of launch. NASA probably ran the number correctly when they were asked this same question. There are numerous other reasons why the auxiliary rockets fire first, but these are advanced topics.