Rocket physics, in the most basic sense, involves the application of Newton's Laws to a system with variable mass. A rocket has variable mass because its mass decreases over time, as a result of its fuel (propellant) burning off.

A rocket obtains thrust by the principle of action and reaction (Newton's third law). As the rocket propellant ignites, it experiences a very large acceleration and exits the back of the rocket (as exhaust) at a very high velocity. This backwards acceleration of the exhaust exerts a "push" force on the rocket in the opposite direction, causing the rocket to accelerate forward. This is the essential principle behind the physics of rockets, and how rockets work.

The equations of motion of a rocket will be derived next.

To find the equations of motion, apply the principle of impulse and momentum to the "system", consisting of rocket and exhaust. In this analysis of the rocket physics we will use Calculus to set up the governing equations. For simplicity, we will assume the rocket is moving in a vacuum, with no gravity, and no air resistance (drag).

To properly analyze the physics, consider the figure below which shows a schematic of a rocket moving in the vertical direction. The two stages, (1) and (2), show the "state" of the system at time

Where:

Note that all velocities are measured with respect to ground (an inertial reference frame).

The sign convention in the vertical direction is as follows: "up" is positive and "down" is negative.

Between (1) and (2), the change in linear momentum in the vertical direction of all the particles in the system, is due to the sum of the external forces in the vertical direction acting on all the particles in the system.

We can express this mathematically using Calculus and the principle of impulse and momentum:

where Σ

Expand the above expression. In the limit as

Since the rocket is moving in a vacuum, with no gravity, and no air resistance (drag), then Σ

The left side of this equation must represent the thrust acting on the rocket, since

Therefore, the thrust

The term

As the rocket loses mass due to the burning of propellant, its acceleration increases (for a given thrust

From equation (2),

which becomes

The mass of the ejected rocket exhaust equals the negative of the mass change of the rocket. Thus,

Therefore,

Again, the term

Integrate the above equation using Calculus. We get

This is a very useful equation coming out of the analysis, shown above. The variables are defined as follows:

In the following discussion we will look at staging and how it can also be used to obtain greater rocket velocity.

Often times in a space mission, a single rocket cannot carry enough propellant (along with the required tankage, structure, valves, engines and so on) to achieve the necessary mass ratio to achieve the desired final orbital velocity (

The figure below illustrates how staging works.

Source: NASA

The picture below shows the separation stage of the Saturn V rocket for the Apollo 11 mission.

Source: NASA

The picture below shows the separation stage of the twin rocket boosters of the Space Shuttle.

Source: NASA

In the following discussion we will look at rocket efficiency and how it relates to modern (non-rocket powered) aircraft.

Rockets can accelerate even when the exhaust relative velocity is moving slower than the rocket (meaning

Rockets can convert most of the chemical energy of the propellant into mechanical energy (as much as 70%). This is the energy that is converted into motion, of both the rocket and the propellant/exhaust. The rest of the chemical energy of the propellant is lost as waste heat. Rockets are designed to lose as little waste heat as possible. This is accomplished by having the exhaust leave the rocket nozzle at as low a temperature as possible. This maximizes the Carnot efficiency, which maximizes the mechanical energy derived from the propellant (and minimizes the thermal energy lost as waste heat). Carnot efficiency applies to rockets because rocket engines are a type of heat engine, converting some of the initial heat energy of the propellant into mechanical work (and losing the remainder as waste heat). Thus, the physics of rockets is related to heat engine physics. For more information see Wikipedia's article on heat engines (opens in new window).

In the following discussion we will derive the equation of motion for rocket flight in the presence of air resistance (drag) and gravity, such as for rockets flying near the earth's surface.

The analysis in this section is similar to the previous one, but we are now including the effect of air resistance (drag) and gravity, which is a necessary inclusion for flight near the earth's surface.

For example, let’s consider a rocket moving straight upward against an atmospheric drag force

where

From equations (1) and (3) and using the fact that acceleration

The sum of the external forces acting on the rocket is the gravity force plus the drag force. Thus, from the above equation,

As a result,

This is the general equation of motion accounting for the presence of air resistance (drag) and gravity. Looking closely at this equation, you can see that it is an application of Σ

Due to the complexity of the drag term

Another important consideration is the design of a rocket that experiences minimal atmospheric drag. At high velocities, air resistance is significant. So for purposes of energy efficiency, it is necessary to minimize the atmospheric drag experienced by the rocket, since energy used to overcome drag is energy that is wasted. To minimize drag, rockets are made as aerodynamic as possible, such as with a pointed nose to better cut through the air, as well as using stabilizing fins at the rear of the rocket (near the exhaust), to help maintain steady orientation during flight.

In the following discussion we will take a closer look at thrust.

The thrust given in equation (3) is valid for an optimal nozzle expansion. This assumes that the exhaust gas flows in an ideal manner through the rocket nozzle. But this is not necessarily true in real-life operation. Therefore, in the following analysis we will develop a thrust equation for non-optimal flow of the exhaust gas through the rocket nozzle.

To set up the analysis consider the basic schematic of a rocket engine, shown below.

Where:

The arrows along the top and sides represent the pressure acting on the wall of the rocket engine (inside and outside). The arrows along the bottom represent the pressure acting on the exhaust gas, at the exit plane.

Gravity and air resistance are ignored in this analysis (their effect can be included separately, as shown in the previous section).

Next, isolate the propellant (plus exhaust) inside the rocket engine. It is useful to do this because it allows us to fully account for the contact force between rocket engine wall and propellant (plus exhaust). This contact force can then be related to the thrust experienced by the rocket, as will be shown. The schematic below shows the isolated propellant (plus exhaust). The dashed blue line (along the top and sides) represents the contact interface between the inside wall of the rocket engine and the propellant (plus exhaust). The dashed black line (along the bottom) represents the exit plane of the exhaust gas, upon which the pressure

where

(Note that, due to geometric symmetry of the rocket engine, the resultant force acts in the vertical direction, and there is no sideways component).

Now, sum all the forces acting on the propellant (plus exhaust) and then apply Newton's second law:

Where:

The right side of the above equation is derived using the same method that was used for deriving equation (1). This is not shown here.

(Note that we are defining "up" as positive and "down" as negative).

Next, isolate the rocket engine, as shown in the schematic below.

where

Now, sum all the forces acting on the rocket engine and then apply Newton's second law:

where

Now, by Newton’s third law

where

Combine the above two equations and we get

Combine equations (5) and (6) and we get

where the term (

Thus, the thrust

This is the most general equation for thrust coming out of the analysis, shown above. The first term on the right is the momentum thrust term, and the last term on the right is the pressure thrust term due to the difference between the nozzle exit pressure and the ambient pressure. In deriving equation (3) we assumed that

A subtle point regarding

The analysis in this section is basically a force and momentum analysis. But to do a complete thrust analysis we would have to look at the thermal and fluid dynamics of the expansion process, as the exhaust gas travels through the rocket nozzle. This analysis (not discussed here) enables one to optimize the engine design plus nozzle geometry such that optimal nozzle expansion is achieved during operation (or as close to it as possible).

The flow of the exhaust gas through the nozzle falls under the category of compressible supersonic flow and its treatment is somewhat complicated. For more information on this see Wikipedia's page on rocket engines (opens in new window). And here is a summary of the rocket thrust equations provided by NASA (opens in new window).

In the following discussion we will look at the energy consumption of a rocket moving through the air at constant velocity.

An interesting question to ask is, how much energy is used to power a rocket during its flight? One way to answer that is to consider the energy use of a rocket moving at constant velocity, such as through the air. Now, in order for the rocket to move at constant velocity the sum of the forces acting on it must equal zero. For purposes of simplicity let's assume the rocket is traveling horizontally against the force of air resistance (drag), and where gravity has no component in the direction of motion. The figure below illustrates this schematically.

The thrust force

Let's say the rocket is moving at a constant horizontal velocity

Thus, we must look at the energy used to push the rocket (in the forward direction) and add it to the energy it takes to push the exhaust (in the backward direction). To do this we can apply the principle of work and energy.

For the exhaust gas:

Where:

For the rocket:

Where:

Set

Therefore,

Now, the time

The total mass of exhaust

Substitute this into equation (8), then combine equations (8) and (9) to find the total mechanical energy used to propel the rocket over a distance

Since

But

Now,

Where:

Substitute the above equation into the previous equation (for

As you can see, the higher the velocity

This tells us that rockets are inefficient for earth-bound travel, due to the effects of air resistance (drag) and the high relative exhaust velocity

For rockets that are launched into space (such as the Space Shuttle) the density of the air decreases as the altitude of the rocket increases. This decrease, combined with the increase in rocket velocity, means that the drag force will reach a maximum at some altitude (typically several kilometers above the surface of the earth). This maximum drag force must be withstood by the rocket body and as such is an important part of rocket analysis and design.

In the following discussion we will look at the energy consumption of a rocket moving through space.

A very useful piece of information in the study of rockets is how much energy a rocket uses for a given increase in speed (delta-

We will need to apply the principle of work and energy to the system (consisting of rocket and propellant/exhaust), to determine the required energy.

The initial kinetic energy of the system is:

Where:

The final kinetic energy of the rocket is:

Where:

The exhaust gases are assumed to continue traveling at the same velocity as they did upon exiting the rocket. Therefore, the final kinetic energy of the exhaust gases is:

Where:

The last term on the right represents an integration, in which you have to sum over all the exhaust particles for the whole burn time.

Therefore, the final kinetic energy of the rocket (plus exhaust) is:

Now, apply the principle of work and energy to all the particles in the system, consisting of rocket and propellant/exhaust:

Substituting the expressions for

Where:

and

and

From equation (4),

where

Hence,

Note that all velocities are measured with respect to an inertial reference frame.

After a lot of algebra and messy integration we find that,

This answer is very nice and compact, and it does not depend on the initial velocity

If we want to find the mechanical power

where

Note that, in the above energy calculations, the rocket does not have to be flying in a straight line. The required energy

This concludes the discussion on rocket physics.

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