Rocket Leaving Earth's Gravitational Field

I stumbled across this seemingly simple question that really stumped me on further thought:

A rocket is intended to leave the Earth's gravitational field. The fuel in its main engine is a little less than necessary, and an auxiliary engine, only capable of operating for a short amount of 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?

My understanding is that to escape the earth's field, there must be sufficient kinetic energy so that the total energy is positive. My first instinct was that, assuming the auxiliary engine operates at a constant power, it will cause the same change in energy regardless of how far it is from the earth, therefore it doesn't matter. However, it seems that where this change in energy occurs is actually important. Can someone help me understand what the correct answer is?

• en.wikipedia.org/wiki/Oberth_effect Commented Apr 6, 2019 at 7:13
• Rockets do not ever leave the earth's gravitational field. Commented Apr 6, 2019 at 9:10
• Would the Oberth effect be relevant here? As I understand it, the Oberth effect assumes that the engine provides a fixed impulse and therefore a fixed change in velocity, and since kinetic energy goes by $v^2$ it will gain the largest amount of energy when it its speed is greatest. But in this case, the fuel provides a set change in energy, so it would not matter where it is when the rocket fires? Commented Apr 6, 2019 at 17:03
• Yes, the Oberth effect is relevant. Remember, fuel doesn't just give KE & momentum to the rocket, it also gives them to the exhaust gases. Commented Apr 8, 2019 at 13:10

In principle, I'll say that it is the same. We can imagine the small auxiliary engine like a little bit of more fuel in the main tank, which is the same as all the other fuel.

In practice, it depends on how efficient the engine is: Rocket engines generate thrust by pushing back the exhaust gases from fuel burn. More efficient engines throw the exhaust faster, thus generating more thrust per unit of fuel. Also the engine dead weight go in the equation.

Fuel is carried by the rocket and is heavy; then the optimal solution will be: - Use the less efficient engine first, so you don't need to carry up all that ineffective fuel & the heavy engine - Use the most weight-efficient engine after.

If you can fire them at the same time, the answer by Paulo Gil works.

It's at the beginning. The way the problem is stated does not forbid the two engines to work simultaneously. As the rocket goes up, all the propellant being transported is gaining potential and kinetic energy. So, the best option is to get rid of propellant (and the auxiliary engine, and its fuel tank) as soon as possible. I'm neglecting the effect of drag forces. If the rocket accelerates more sooner, drag will be larger because the atmosphere is denser at the bottom and speed will be higher in regions where drag is relevant, but drag loss is typically much smaller than gravity effects, and the eventual advantage of getting rid of the engine and tanks should more than compensate.

$$\def\half{{\textstyle {1 \over 2}}}$$ Neither. Choose an instant of time $$t$$ and put yourself in the rocket's rest frame at that instant. Auxiliary engine will impart rocket some velocity $$v$$, irrespective of the instant chosen. If $$u$$ is rocket's velocity wrt Earth at $$t_-$$, its velocity at $$t_+$$ will be $$u+v$$ and the increment in energy will be $$\half\,m\,(u + v)^2 -\half\,m\,u^2 = \half\,m\,v\,(2\,u + v).$$ This is maximum when $$u$$ is maximum, i.e. at the instant when main fuel is exhausted.

The above argument didn't take into account rocket's mass variation as main fuel gets consumed. The effect of mass decrement is of increasing $$v$$ so that my conclusion is strengthened.

Theory simply suggests that you may use the auxillary engine whenever you want because it is going to provide the same output and power so the amount by which it increases the energy of the system is the same. However from practical and engineering aspects it would be the best to use it up first because if used first it won't be needed further wich will help in reducing mass of the system to make it more effective.