A rocket is intended to leave the Earth's 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
When the rocket has nearly stopped with respect to the Earth
It doesn't matter
Can't say
A.
at take-off
The rocket has to reach the highest possible total energy. If the zero level of gravitational potential energy is infinitely far away, then the energy of the rocket standing on the surface on the earth is negative. The energy released during the operation of the engines increases the total energy of the rocket, and the rocket can leave the earth's gravitational field if the sum of its potential and kinetic energies becomes positive.
The energy released in the course of the operation of the principal and auxiliary engines increase the total energy of the rocket and its ejected combustion products by a fixed value; this increase is independent of the moment when the engines are scratched on. However, the speed at which the combustion products fall back to the earth does depend on the timing of the rocket's operations. Indeed, if the auxiliary engine starts working when the rocket is at a greater height, the combustion fall further and their speed and total energy are higher when they hit the ground. This means that the sooner the auxiliary engine is switched on. the higher, the energy ultimately acquired by the rocket. The same concept is valid for the principal engine, and if the only energy consideration applies, it is best to operate the engines for the shortest time and at the highest thrust.
