Active Structures: Difference between revisions

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==Mass Streams==
==Mass Streams==
<div style="border-radius: 10px; background-color:#ffd966;">
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|rowspan="2" style="text-align:left; background-color:#ffd966; color:#d48200; padding-right:15px"|<div style="font-size: 50px; text-align:center;"> ! </div>
|style="background-color:#ffd966; color:black; text-align:left; padding-left:10px; font-size: 20px;"|'''Notice:'''
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|border="0" style="border-collapse:collapse; background-color:#ffd966; color:black; padding-left:10px; width: 99%; text-align:left; padding-bottom:10px;"| This section is currently a work in progress and information here may not be correct.
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</div>
Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.
Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.


The amount of kinetic energy (KE) required for each pellet or particle for the accelerator in the stream is given by:
Before we can calculate the kinetic energy required for each particle/pellet in the stream, important in determining energy input, the force, or the weight of the active structure must be calculated:


<math alt=> E=A_gh(m+M)/N </math>
<math alt=> F=MA_g</math>
* Where <math alt=>E</math> is the amount of KE required for each pellet or particle in the stream.
* Where <math alt=>m</math> is the total mass of the stream.
* Where <math alt=>h</math> is the altitude of the deflector.
* Where <math alt=>M</math> is the mass of the structure.
* Where <math alt=>N</math> is the number of particles or pellets in the stream.


The total KE must be greater than the gravitational potential energy of the stream in order to support any amount of mass. Assuming perfectly lossless deflectors and there is no other mechanism to which energy can be lost there should be no further additional input of energy.
* Where F is the weight of the active structure.
* Where M is the mass of the active structure.


: Calculated example:
With that information in mind, the kinetic energy of the stream is given by:


: Given a 1,000 metric ton structure on Earth, a height of 10 km, a total circulating mass of 1 metric ton, and 1000 pellets in the stream, the required KE for the pellet is 98.2 megajoules. Considering that there are 1000 pellets, the mass of the pellet is 1 kg and the velocity of the pellet is 14 km/s<ref>Rearranging the mass-velocity KE equation <math alt=>KE=1/2mv^2</math> to solve for velocity gives <math alt=>v=\sqrt{2KE}/\sqrt{m}</math>. Plugging in the variables, 98.2 megajoules and 1 kg gives the velocity, 14 km/s.</ref>.
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Long form</div>
<div class="mw-collapsible-content">
<math alt=> KE_s = (v-\sqrt{v^2-2A_gh} )/A_g \cdot F/(2 \sqrt{v^2-2A_gh}) v^2 </math>
</div></div>


-----
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Simple form</div>
<div class="mw-collapsible-content">
<math alt=> KE_s = hF(b(\frac{1}{\sqrt{1-1/b }}-1)) </math>
<br/><math alt=> b = v^2/2A_gh</math>
</div></div>


Before we can calculate the kinetic energy required for each particle/pellet in the stream, important in determining energy input, the force, or the weight of the active structure must be calculated:
* Where <math alt=>KE_s</math> is the kinetic energy of the stream.
* Where <math alt=>v</math> is the velocity of the stream.
* Where <math alt=>h</math> is the altitude of the deflector.


<math alt=> F=MA_g</math>
Notes:
# As a rule of thumb, whenever <math alt=>v>\sqrt{10A_gh}</math> energy costs are almost optimal. Provided that the stream velocity is above that threshold, you can forego the complex equations involved and go with <math alt=> KE_s \approx 0.6hF</math> instead. The choice of pellet characteristics can then fall under engineering constraints.
# The functions overestimate the energy content due to the presence of a constant gravitational field - in reality, it would not be constant.


* Where F is the weight of the active structure.
Finally, the kinetic energy of the stream is divided by the number of pellets/particles in the stream to give the kinetic energy of each pellet/particle:
* Where M is the mass of the active structure.
 
With that information in mind, the kinetic energy of the particle/pellet is given by:


<math alt=> KE = (v-\sqrt{v^2-2A_gh} )/A_g \cdot F/(2 \sqrt{v^2-2A_gh}) v^2 </math>
<math alt=>KE_p=KE_s/N</math>


* Where <math alt=>KE</math> is the kinetic energy of the particle/pellet.
* Where <math alt=>KE_p</math> is the kinetic energy of the particle/pellet in the stream.
* Where <math alt=>v</math> is the velocity of the particle/pellet.
* Where <math alt=>N</math> is the number of pellets/particles in the stream.
* Where <math alt=>h</math> is the altitude of the deflector.


The mass of the particle/pellet required can be calculated with:
Additionally, the mass of the pellet/particle can be calculated with:


<math alt=>m=\sqrt{2KE}/v^2</math>
<math alt=>m=2KE_p/v^2</math>


* Where <math alt=>m</math> is mass of the particle/pellet.
* Where <math alt=>m</math> is mass of the particle/pellet.
Line 74: Line 70:
A calculated example:
A calculated example:


: Given a 1,000 ton structure on Earth (implying a <math alt=>A_g</math> of 9.8066 m/s<sup>2</sup>, a height of 10 km and pellet velocity of 90 km/s, the force is 9,806,650 newtons and the kinetic energy of the pellet is 49,034.14 megajoules. Calculating the mass with the KE and given velocity we get: 12.107 kg.
: Given a 1,000 ton structure on Earth (implying a <math alt=>A_g</math> of 9.8066 m/s<sup>2</sup>), a height of 10 km, 1,000 pellets in the stream and pellet velocity of 10 km/s, the force is 9,806,650 newtons and the kinetic energy of the pellet is 49,105.495 megajoules. Secondarily calculating the mass with the KE divided by the number of pellets and given velocity we get 0.982 kg.


=Active Structures=
=Active Structures=
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<references group="GAE"/>
<references group="GAE"/>


==Derivation of the Particle/Pellet Kinetic Energy Equation==
==Derivation of the Stream Kinetic Energy Equations==
# Given that the equation is solving for kinetic energy and involves gravity, it follows that the equation must also involve gravitational potential energy (GPE).  
===Long Form===
# In this case, the GPE is the GPE of the structure and since there is an opposite acceleration for active support, it means that there is a difference in the equation involved.
# Given a classical particle accelerating upwards, it will achieve a final velocity of <math alt=>v_f=\sqrt{v^2-2A_gh}</math> where v is the initial speed.<ref group="SKE">https://www.school-for-champions.com/science/energy_potential.htm <br/> The expression is derived from the expression for the velocity of falling objects: <math alt=>V=\sqrt{2A_gh}</math>. This can be thought as of the amount of velocity that gravity subtracts from the initial velocity.</ref>.
# The difference is between the KE and the GPE of the structure; the difference divided by gravity and height gives the mass supported by the structure.
# The necessary force for deflecting the particle down is given by <math alt=>F=2\dot{m}v_f</math><ref group="SKE"> <math alt=>\dot{m}</math> is the mass of the particle <math alt=>\bullet</math> the repetition rate, the repetition rate being analogous to <math alt=>N</math>, and <math alt=>\dot{m}</math> analogous to <math alt=>KE_s/N</math>.</ref>. The total energy content is then <math alt=>mv^2/2</math>.
# This gives the expression: <math alt=>M=(E-mgh)/g/h</math> where <math alt=>M</math> is the mass supported by the structure, <math alt=>E</math> is the total KE of the mass stream, <math alt=>m</math> is the mass of the stream, <math alt=>g</math> is the gravitational acceleration and <math alt=>h</math> is the relevant portion of the height of the structure, in this case the altitude of the deflector.
# The stream must be long enough for one full circulation up and down, so any particle going up the height of the stream to the deflector takes a time <math alt=>T</math> where <math alt=>v_f = v - A_gT \Rightarrow \color{green}{T = (v-v_f)/A_g}</math>, for a full circulation it takes <math alt=>2T</math>.
# Rearranging the expression to solve for energy and dividing by the number of pellets/particles in the stream gives the final equation.
# Therefore <math alt=>m = 2T\dot{m}</math>, so <math alt=> KE_s = T\dot{m}v^2</math>. To get rid of <math alt=>\dot{m}</math> we divide <math alt=>F</math> by <math alt=>2v_f</math>.
# This gives our stream kinetic energy equation, taking the limit as <math alt=>v</math> monotonically approaches infinity <math alt=>hF/2</math> from above.
 
===Simple Form===
# The simple form is just a factorized version of the long form, where <math alt=>k</math> is defined as <math alt=>2A_gh/v^2</math> or <math alt=>1/b</math>.
 
===Regarding the notes===
# The simple form shows that whenever <math alt=>v^2/2A_gh > 5</math>, the energy content in the stream is less than <math alt=>0.6hF</math>. Rearranging the equation for <math alt=>v</math> and accounting for 5 gives the rule of thumb.
# The upper bound nature of the equations means that:
:: For <math alt=>v > \sqrt{10A_gh}</math>, <math alt=>0.5hF < E < 0.6hF</math> for any <math alt=>h</math> and any gravitational field that is decreasing.
 
----
'''Reference for the Derivation of the Stream Kinetic Energy Equations'''
<references group="SKE"/>


=Credit=
=Credit=
Line 136: Line 145:
* To AdAstraGames for contributing useful information and sweeping the article.
* To AdAstraGames for contributing useful information and sweeping the article.
* To Sevoris for writing the safety engineering section.
* To Sevoris for writing the safety engineering section.
* To MatterbeamToughSF, Kerr and Favalli for help with the math.
* To MatterbeamToughSF, Kerr in particular and Favalli for help with the math.
 
[[Category:Transportation & Infrastructure]][[Category:Infrastructure]][[Category:Structures]][[Category:Physics & Math & Engineering‏‎]]

Latest revision as of 13:25, 23 April 2024

Active structures rely on constant power input, in addition to the material and mechanical properties of their construction materials (active support). This is in contrast to passive structures, which solely rely on the aforementioned properties (passive support). An example of an active structure is the force of a jet of water holding up a tethered lid of a trashcan in the air, versus the passive structure of a concrete pillar.

Nearly everything, from skyscrapers to houses are passive structures. Low-power active structures are in use now, for things like roof support.

The advantage of active structures is that they can be much more massive than passive structures [Footnote 1], enabling structures many kilometers tall without requiring significant tapering. Some proposals for non-rocket launch infrastructure rely on active support, with the advantage of the option for being built by modern, existing materials.

Most known designs of active structures rely on the force of a stream of mass to support them, using an accelerator to drive the mass stream.

  1. ↑ Passive structures can attain extremely tall heights, however, they require pyramid-like tapering with a significant base area to support the weight.

Active Support Principles

As gravity[1]is what pulls down objects, active support must counteract gravity. Since it is the acceleration that causes objects to be pulled down, it follows that active support should accelerate in the opposite direction; the acceleration must be equal to gravity to support the structure.

The gravitational acceleration of the planet is given by:

  • Where is the gravitational acceleration.
  • Where is the universal gravitational constant, defined to be 6.6743e11 m3/kg/s2.
  • Where is the mass of the planet.
  • Where is the radius of the planet.

On Earth, equals 9.80665 m/s2, a constant known as .

Mass Streams

Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.

Before we can calculate the kinetic energy required for each particle/pellet in the stream, important in determining energy input, the force, or the weight of the active structure must be calculated:

  • Where F is the weight of the active structure.
  • Where M is the mass of the active structure.

With that information in mind, the kinetic energy of the stream is given by:

Long form

Simple form


  • Where is the kinetic energy of the stream.
  • Where is the velocity of the stream.
  • Where is the altitude of the deflector.

Notes:

  1. As a rule of thumb, whenever energy costs are almost optimal. Provided that the stream velocity is above that threshold, you can forego the complex equations involved and go with instead. The choice of pellet characteristics can then fall under engineering constraints.
  2. The functions overestimate the energy content due to the presence of a constant gravitational field - in reality, it would not be constant.

Finally, the kinetic energy of the stream is divided by the number of pellets/particles in the stream to give the kinetic energy of each pellet/particle:

  • Where is the kinetic energy of the particle/pellet in the stream.
  • Where is the number of pellets/particles in the stream.

Additionally, the mass of the pellet/particle can be calculated with:

  • Where is mass of the particle/pellet.

A calculated example:

Given a 1,000 ton structure on Earth (implying a of 9.8066 m/s2), a height of 10 km, 1,000 pellets in the stream and pellet velocity of 10 km/s, the force is 9,806,650 newtons and the kinetic energy of the pellet is 49,105.495 megajoules. Secondarily calculating the mass with the KE divided by the number of pellets and given velocity we get 0.982 kg.

Active Structures

Existing

  • The air-supported fabric roofs of the Tokyo Dome, Japan, and the Silverdome, USA use (and for the latter, used) constant fan pressure to keep the roofs aloft.

Proposed

  • The Lofstrom Launch Loop is a thin 2000+ km long and 80 km tall active structure, and uses its own mass stream to help launch payloads to orbit. It uses attractive magnetic levitation for the mass stream; the mass stream is a solid continuous iron rotor. The loop suffers from some unaddressed instability concerns.
  • The Space Cable is a similar concept to the launch loop. It differs from the launch loop in that it uses magnetically interacting bolts instead of a continuous rotor, is smaller in length, and has addressed instability concerns.
  • The Orbital Ring, which uses a mass stream travelling faster than the orbital velocity to support a ring above a planet, as the stream keeps it from falling through momentum, and is tethered to the earth for stability.
  • The Pneumatic Freestanding Tower, which uses pressurized gas to support large structures such as a space tower. It utilizes compressors to provide pressurized gas and alleviate leaks. The main concerns are buckling due to the height of the tower, though it has mechanisms in place to prevent this.

Control Systems

Control Systems
Active structures can suffer from stability issues as mentioned before, such as for example in the launch loop unstable attractive magnetic levitation of the mass-stream in the launch-loop requiring active control of the deflector magnet. The unpredictable winds in the atmosphere are also a concern. Control systems are also needed in even just skyscrapers, with devices like tuned mass dampers to deal with vibration[2].

Safety Engineering

Active structures are subject to the problem of how to ensure that they don't fail, or a bit worse, only fail gracefully, when something in their active systems breaks down. This is not a question of if; entropy breaks everything. All electrical and mechanical systems have a mean time between failure. If an active structure is only supported by a single active support "string", the failure of that "string" will cause a catastrophic failure.

By adding redundancy to our structure, we can ensure it can tolerate the failure of some of its components, and possible "fail gracefully" with a time period allowing for evacuation and response measures to be taken, and/or a "controlled failure" of the structure in which terminal velocity of the falling structure and the production of energetic debris is reduced.

Accepting an increase in mass, we split the support power required between some "strings" operating in parallel. Each string only operates at a partial power, with an oversize factor of added on top. If one or a few of the strings in the parallel system fail, the other strings are ramped to full power, generating sufficient support power to ensure the active structure remains standing despite the failure of some of its "strings". We can also use this to shut down strings intentionally for inspections, maintenance, overhauls, or other work, overall allowing us also to keep the active structure alive over time by incrementally replacing and upgrading its parts.

Usually safety redundancies have at least three systems operating in parallel. You may consider having a larger number of systems.

Note: in terms of safety engineering, no degree of redundancy reduces the chance of a total, catastrophic failure to zero. There is some chance that even a very redundant system may experience the failure of all its critical components at once. But this is given for any system, and you could consider pushing the safety factor of an active structure to the same point (or beyond) any passive structure.

Additional Reading

Additional References

  1. ↑ https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
    Wikipedia article about gravity in classical mechanics.
  2. ↑ https://en.wikipedia.org/wiki/Tuned_mass_damper
    https://en.wikipedia.org/wiki/Active_structure
    https://www.youtube.com/watch?v=f1U4SAgy60c
    Wikipedia articles about control systems and the aforementioned tuned mass damper, as well a Practical Engineering video on it.

Derivation of the Gravitational Acceleration Equation

  1. The gravitational force equation[GAE 1] is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = G(m_1m_1/r^2)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is the force, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is the universal gravitational constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} is the mass of the first object, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2} is the mass of the second object and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is the distance between their centers of mass.
  2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} becomes the radius of the planet from the frame of reference of a planet.
  3. The equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F = MA} , which gives the force needed to accelerate an object is rearranged to give acceleration, thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=F/M}
  4. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A =F/M} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is the acceleration, divide by the mass of the second object and therefore cancel out its inclusion in the expression, giving you the gravitational acceleration equation.

Reference for the Derivation of the Gravitational Acceleration Equation

  1. ↑ https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
    Reference for the gravitational force equation.

Derivation of the Stream Kinetic Energy Equations

Long Form

  1. Given a classical particle accelerating upwards, it will achieve a final velocity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_f=\sqrt{v^2-2A_gh}} where v is the initial speed.[SKE 1].
  2. The necessary force for deflecting the particle down is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F=2\dot{m}v_f} [SKE 2]. The total energy content is then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle mv^2/2} .
  3. The stream must be long enough for one full circulation up and down, so any particle going up the height of the stream to the deflector takes a time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_f = v - A_gT \Rightarrow \color{green}{T = (v-v_f)/A_g}} , for a full circulation it takes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2T} .
  4. Therefore Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m = 2T\dot{m}} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle KE_s = T\dot{m}v^2} . To get rid of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{m}} we divide Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2v_f} .
  5. This gives our stream kinetic energy equation, taking the limit as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} monotonically approaches infinity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle hF/2} from above.

Simple Form

  1. The simple form is just a factorized version of the long form, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2A_gh/v^2} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/b} .

Regarding the notes

  1. The simple form shows that whenever Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v^2/2A_gh > 5} , the energy content in the stream is less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.6hF} . Rearranging the equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} and accounting for 5 gives the rule of thumb.
  2. The upper bound nature of the equations means that:
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v > \sqrt{10A_gh}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.5hF < E < 0.6hF} for any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} and any gravitational field that is decreasing.

Reference for the Derivation of the Stream Kinetic Energy Equations

  1. ↑ https://www.school-for-champions.com/science/energy_potential.htm
    The expression is derived from the expression for the velocity of falling objects: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V=\sqrt{2A_gh}} . This can be thought as of the amount of velocity that gravity subtracts from the initial velocity.
  2. ↑ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{m}} is the mass of the particle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bullet} the repetition rate, the repetition rate being analogous to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot{m}} analogous to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle KE_s/N} .

Credit

To Tshhmon for writing the article

  • To SOPHONT SIMP and pMXoTJFu for sweeping the article.
  • To AdAstraGames for contributing useful information and sweeping the article.
  • To Sevoris for writing the safety engineering section.
  • To MatterbeamToughSF, Kerr in particular and Favalli for help with the math.
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