# Rocket equation: is there an optimal thrust?

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• Carlos Peres
In summary, the conversation discusses rocket equations and optimal thrust for maximum delta V when launching from the ground, taking into account factors such as mass ejection rate, acceleration, and atmospheric drag. It is determined that a higher thrust is better, but it must also be balanced with the ability of the rocket's structure and cargo to handle the acceleration. Throttling back the thrust is necessary for hovering, but it is not a common practice due to its inefficiency. The direction of the thrust is also important, with perpendicular thrust to the force of gravity being most efficient, and taking advantage of the Oberth effect can also increase efficiency.
Carlos Peres
This is my first post. I looked through many threads and could not find an answer to the question below.

Well known equations in rocketry are:

Delta Velocity = Ejection Velocity * ln (Final Mass / Initial Mass )

and

Net Thrust = Mass Ejection Rate * Ejection Velocity – Current Mass * Acceleration

The first equation does not refer to Mass Ejection Rate. If the Mass Ejection Rate (MER) is too small the thrust will not be sufficient for the rocket to take off. Increasing the MER one might get:

Rocket takes off once Thrust > Weight

Rocket takes off immediately at low acceleration

Rocket takes off immediately at high accelerationQuestions: Ignoring drag, is there an optimal thrust that provides maximum delta V for a rocket taking off from the ground?

Carlos Peres said:
Questions: Ignoring drag, is there an optimal thrust that provides maximum delta V for a rocket taking off from the ground?
The faster the burn and the higher the thrust, the better... as long as the structure and the cargo can handle the acceleration.

One way of seeing this is to imagine what happens if the thrust is no greater than the weight of the rocket: it hovers, gaining no altitude, until the fuel is exhausted and it falls to the ground.

sophiecentaur
Nugatory said:
as long as the structure and the cargo can handle the acceleration
For a launch from within an atmosphere, one does not want to get too fast too low. Your savings from spending less time in the gravity well could be offset by losses from spending more energy to air resistance. This is in addition worrying about whether the craft can handle the stresses from air resistance. For instance, see https://en.wikipedia.org/wiki/Max_Q

Thank you, Nugatory and jbriggs44.
Yes, that is why when a Falcon 9 is launched at about 1 minute of flight and 12 Km of altitude it is throttled down temporarily while going through maximum dynamic pressure. Once atmospheric pressure is reduced is is throttled back to full power.
Still, ignoring drag and material strength, is there an optimal thrust for maximum delta V?

Carlos Peres said:
Still, ignoring drag and material strength, is there an optimal thrust for maximum delta V?
Ignoring drag and material strength @Nugatory has provided the answer in post #2 above. The faster the better.

sophiecentaur
Hi Nuggatory
Tsiolkowski's equation is considered to be THE rocket equation. The higher the thrust the better is a constraint not implied by the equation.

Thanks jbriggs444
After I posted my response above I realized something you said did not quite fit. At constant thrust a rocket cannot hover as its mass is being decreased. Right?

Sorry kbriggs444. the hovering statement also came from Nugatory.

Carlos Peres said:
Hi Nuggatory
Tsiolkowski's equation is considered to be THE rocket equation. The higher the thrust the better is a constraint not implied by the equation.
If you are launching from within a gravity well, the more time you spend in the well, the more delta-V you lose to the well. Yes, Tsiolkovski gives you a fixed delta V regardless of mass flow rate. But you can budget that delta-V toward hovering in place or toward escaping the gravity well by choosing your mass flow rate.

Edit: I look at a gravity field as if it were a leak in your tank of "delta V". The more time you spend leaking delta V, the more you'll lose. The closer you are to the planet you're launching from, the stronger the gravity and the faster you're leaking delta V.

A high mass flow rate means that you burn the fuel lower in the well and you spend less time subject to the stronger gravity there. The faster you burn, the better.
After I posted my response above I realized something you said did not quite fit. At constant thrust a rocket cannot hover as its mass is being decreased. Right?
@Nugatory spoke of a rocket using just enough thrust to hover. Yes, the required thrust would not be constant but would decrease as the fuel left on board depletes.

Hi jbrigss444
You are not implying that the thrust would decrease as fuel on board depletes, are you? It takes some smart technology to make a rocket hover.

Carlos Peres said:
Hi jbrigss444
You are not implying that the thrust would decrease as fuel on board depletes, are you? It takes some smart technology to make a rocket hover.
If you want to hover, you would need to throttle back. Yes.

Edit: And yes, nobody hovers on rocket engines because it is such an obvious waste of fuel.

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There's also an optimum direction. Once beyond significant effects from the atmosphere, the closer the thrust is to being perpendicular to the force of gravity, the more efficient. If the thrust is perpendicular to the force of gravity, then all of the thrust goes into increasing velocity.

update - willem2 notes in his post that thrust in the same direction as velocity is most efficient. In the case of a circular orbit, the velocity is perpendicular to gravity, but in the case of an elliptical orbit, the velocity is not perpendicular to gravity. jbrigs444 notes the Oberth effect, where it's most efficient to apply thrust at the perigee (fastest speed) of an orbit for the greatest change in kinetic energy, and this coincides with the velocity being close to perpendicular to gravity.

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rcgldr said:
There's also an optimum direction. Once beyond significant effects from the atmosphere, the closer the thrust is to being perpendicular to the force of gravity, the more efficient. If the thrust is perpendicular to the force of gravity, then all of the thrust goes into increasing velocity.
If it were not for the pesky planetary surface getting in the way, one could allow the trajectory to dip below the starting point and make use of the Oberth effect.

rcgldr said:
There's also an optimum direction. Once beyond significant effects from the atmosphere, the closer the thrust is to being perpendicular to the force of gravity, the more efficient. If the thrust is perpendicular to the force of gravity, then all of the thrust goes into increasing velocity.
The thrust is most efficient if it is in the same direction as the velocity. That will increase the kinetic energy the most. The only reason you won't use that is if you will crash into the earth.
Thrust perpendicular to gravity does not always do this. For example if you thrust sideways in a circular orbit, you will thrust perpendicular to gravity, but you will remain in a circular orbit, only the plane of the orbit will change.

jbriggs444
willem2 said:
The thrust is most efficient if it is in the same direction as the velocity.
Note, however, that this is only a local optimum. If you want to increase the energy of the craft with a single impulsive burn applied immediately, the optimal direction is indeed parallel to the velocity.

If you want to increase the energy but you have the option of performing multiple burns and if you are in a gravity well then a global optimum might involve firing anti-parallel [and at apogee] to produce a grazing orbit and then firing parallel at perigee to obtain the maximum energy gain.

If one is at rest on the surface of a rotating spherical body devoid of atmosphere then the local optimum is the same as the global optimum and we are all in agreement: "fire all your engines at once [horizontally] and explode into space"

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rcgldr said:
There's also an optimum direction. Once beyond significant effects from the atmosphere, the closer the thrust is to being perpendicular to the force of gravity, the more efficient. If the thrust is perpendicular to the force of gravity, then all of the thrust goes into increasing velocity.

willem2 said:
The thrust is most efficient if it is in the same direction as the velocity.
True, but I was thinking of the Oberth effect as posted by jbriggs444. The thrust is most efficient at perigee (fastest speed) of an elliptical orbit, where the velocity is close to being perpendicular to gravity. If the goal is to transition from a smaller circular orbit to a larger circular orbit, then for a Hohmann transfer orbit, the two burns are done at perigee and apogee, when velocity is close to perpendicular to gravity.

https://en.wikipedia.org/wiki/Hohmann_transfer_orbit

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## 1. What is the rocket equation and why is it important?

The rocket equation, also known as the Tsiolkovsky rocket equation, is a mathematical equation that describes the motion of a rocket in terms of its mass, velocity, and the amount of propellant it carries. It is important because it allows scientists and engineers to calculate the change in velocity, or delta-v, that a rocket can achieve based on its mass and the amount of propellant it carries. This is crucial in designing and optimizing rocket launches.

## 2. How is thrust related to the rocket equation?

Thrust is a key component of the rocket equation as it is the force that propels the rocket forward. The rocket equation states that the change in velocity of a rocket is directly proportional to its thrust and the natural logarithm of its initial and final mass. This means that increasing the thrust of a rocket can greatly impact its overall performance.

## 3. Is there an optimal thrust for a rocket?

Yes, there is an optimal thrust for a rocket. This is because as the thrust increases, so does the amount of propellant needed to achieve a certain change in velocity. However, increasing thrust also means increasing the weight of the rocket, making it less efficient. Engineers must balance these factors to determine the optimal thrust for a specific rocket and mission.

## 4. Can the rocket equation be applied to all types of rockets?

Yes, the rocket equation can be applied to all types of rockets, including chemical, nuclear, and electric. However, the specific parameters and variables may differ depending on the type of rocket and its propulsion system. For example, an electric rocket may have a much lower thrust than a chemical rocket, but it can still be calculated using the rocket equation.

## 5. How accurate is the rocket equation?

The rocket equation is a highly accurate mathematical model for calculating the motion of rockets. However, it does not account for external factors such as air resistance and gravity, which can affect the performance of a rocket. In addition, real-world rockets may deviate slightly from the predicted values due to variations in engine performance and other factors. Nevertheless, the rocket equation is a valuable tool for designing and optimizing rocket launches.

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