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The various types of collisions, NASA’s EmDrive And Generalized Form of Newton’s
Third Law of motion
Ajay Sharma
Fundamental Physics Society His Mercy Enclave, Post Box 107 GPO Shimla 171001 HP India
Email: [email protected]
In Newton’s Principia the Third Law of Motion states: ‘to every action there is always opposed
an equal reaction; or the mutual actions of two bodies upon each other are always equal, and
directed to contrary parts.’ The Law only refers to the ‘bodies’, not other factors. The Law does
not take into account the inherent characteristics, nature, compositions, flexibility, rigidity,
magnitude, elasticity, distinctiveness of interacting bodies etc. The bodies may be of steel, wood,
rubber, cloth, wool, sponge, spring, typical plastic, porous material, specifically fabricated
material etc. The interacting bodies may be solid, liquid gas or mixture of all. For all such
bodies if the action is same, then the reaction must be the same. In the qualitative explanation
(without any mathematical equation) given after the definition to the Law, Newton expressed
Action and Reaction in terms of push or pull (force) and motion (velocity). In numerous cases
we find that action and reaction are not equal as the Principia’s Third Law of Motion as it does
not take in account vital factors like inherent nature and characteristics of interacting bodies.
Thus, to take elusive and effective factors into account, the Third Law of Motion is generalized
as: ‘To every action there is always an opposed reaction, which may or may not be equal in
magnitude, depending upon the inherent characteristics of the process.’ Mathematically, Action
=-K Reaction, the value of K may be equal to, less than, or greater than unity. Newton did not
give The Third Law of Motion in proportionality form like preceding Second Law Of Motion of
the Principia.
The value of coefficient of proportionality K takes in account the inherent
characteristics of the process. Newton stated the law in terms of action and reaction which even
now are devoid of units and dimensions. Now action and reaction should be replaced by the
physical quantities having units and dimensions in The Third Law Motion and interpreted in
terms of specific equation. NASA’s recent EmDrive confirms that there is no reaction for
definite action, thus confirming the inadequacy of third law. It is consistent with generalized
form of the law .
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1.0 The Principia’s Third Law of Motion
The original form of the Third Law of Motion is:
To every action there is always opposed an equal reaction; or the mutual actions of two
bodies upon each other are always equal, and directed to contrary parts. [1,2]
Action = - Reaction
or Mutual action of one body = - Mutual reaction of other body
(1)
(1)
The equation indicates the action and reaction are universally equal without constraints.
In Newton’s time it was beginning of science, thus physical quantities, units, dimensions were
not defined. Thus the terms action and reaction do not possess units and dimensions. However in
the qualitative explanation given after the definition to the law, Newton expressed action and
reaction in terms of push or pull (force) and motion (velocity), in the Principia at page 20.
Newton did not write eq. (1) in the Principia. Newton wrote the Principia’s Second Law of
Motion in proportionality form like Law of Gravitation, but The Third Law is expressed in
equality form. The detailed analysis reveals that we may understand Newton’s Third Law of
Motion as conservation of kinetic energy as bodies move. The detailed analysis reveals that we
may understand Newton’s Third Law of Motion as conservation of kinetic energy in some cases.
However kinetic energy may change from one form to other, and various types of energy are
inter-convertible. Action and reaction do not have units and dimensions.
2.0 Newton’s Original Explanation of the Third Law of Motion in the Principia
“Whatever draws or presses another is as much drawn or pressed by that other. If you press a
stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a
rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended
rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the
stone as it does the stone towards the horse, and will obstruct the progress of the one as much as
it advances that of the other.
If a body impinges upon another and by its force change the motion of the other, that body also
(because of the quality of, the mutual pressure) will undergo an equal change, in its own motion,
towards the contrary part. The changes made by these actions are equal, not in the velocities but
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in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments.
For, because the motions are equally changed, the changes of the velocities made towards
contrary parts are reciprocally proportional to the bodies. This law takes place also in
attractions, as will be proved in the next scholium.”
Thus in the short description of the law Newton did not give any equation
to measure or calculate the magnitudes of action and reaction. Thus Newton provided
conceptual, thematic and philosophical explanation of the law, not quantitative. Newton’s law is
applicable to all interacting bodies, action and reaction are always equal and opposite for all
interacting bodies. It can also be considered applicable for waves. The definition of the law does
not restrict its applications i.e. it is universally applicable for all types of bodies may be elastic or
non-elastic bodies etc. Now law has been interpreted in those experiments, which are not
critically and quantitatively discussed so far.
Thus law does not consider the inherent characteristics, nature, composition,
flexibility, rigidity, magnitude, distinctiveness of interacting bodies etc. There is no factor which
accounts for the above significant factors. The bodies may be composed of steel, wood, rubber,
cloth, wool, sponge, spring, elastic , plastic, typical plastic, porous material, especially
fabricated etc. The bodies may be solids, liquids or gases or mixture of all. For all such bodies if
the action is the same, then the reaction must be the same. Some bodies may have inherent
tendencies to restore original shape when deformed e.g. rubber or spring bodies. Whereas others
do not show any such tendency e.g. mud or flour balls or chewing gum. Furthermore, in
collisions comparative size and point of impact, of target and projectile, roughness of surfaces
and resistive forces play significant roles. Thus law may be regarded as under ideal conditions
only.
These are very significant factors affecting the results and are taken in account via a coefficient
of proportionality in the generalized law. In the generalized or extended form of the law, a
coefficient of proportionality comes in equation, which takes all elusive factors into account.
Further Newton’s law is completely silent about electrical and magnetic interactions between the
various bodies, these are exception in some cases.
Consequently Newton’s third law is generalized, so that the law becomes complete taking all
factors in account.
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3.0
Newton did not consider the following experiments.
(i) Consider two exactly similar rubber balls of mass 0.1 kg each. Let both balls are dropped
from height of 1 meter, on the floor, each will have action ( W=0.98N).
After striking the floor the both rubber balls rebound identically i.e. reach the same point (1m
above ) in the same time , thus reaction forces must be the same. Thus
Action = -Reaction
This observation is completely consistent with Newton’s Third Law of Motion. But there are
numerous other examples as well. The law cannot be established just on the basis of one
observation.
(ii) If all other parameters remain the same (rubber ball 0.1kg, height 1 meters ) but a
mattress is kept over the floor. In the identical experiment we find that the rubber ball rebounds
to half meter. The reason is that a part of kinetic energy of ball is absorbed by mattress in form
of potential energy, heat energy, sound energy etc. thus ball experiences reduced reaction than
previous case ( mattress is not placed over the surface). Thus cloth ball rebounds to the lesser
height. Hence
Action = ½ Reaction
So in this case, Action and Reaction are not equal, as inherent nature and characteristics of floor
and mattress are different than floor.
(iii) Similar results can be obtained if one rubber ball and one cloth ball (same action, weight =
0.98N ) are dropped from height 1m. The cloth balls rebounds slowly and reach height less than
1m. The reason is the kinetic energy of the cloth ball is converted to its potential energy, heat
energy, sound energy etc. The reaction is smaller in this case compared to rubber ball. Thus
rebound velocity of ball is less than incident velocity and ball rebounds to the lesser height.
Reaction <Action
(v) Further, if a typical sponge ball of mass 0.1kg (W=0.98N, Action) is dropped from the
height 1m. Then it does not rebound at all. Thus
Action = 0.98N
Reaction =0
Again action and reaction are not equal.
(vi) If a stone of mass 0.1kg (W=0.98N, Action) is dropped on the stretched paper, then it pierce
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through it. The reason is that the potential energy of paper ( inherent nature and characteristics)
is too less to cause reaction to stop stone which has significant kinetic energy.
(v) Super elastic collisions: These are collisions in which velocity (KE) after collision is more
than velocity (KE) before the collisions. It is possible with specially fabricated projectile and
target.
In super elastic collisions the potential energy is converted into kinetic energy so that
the total kinetic energy of the colliding objects is greater after the collision than before. Imagine
a massive spring of high spring constant being compressed with an extremely delicate device
(i.e. once touched will release the spring). Now we can envision that a collision between a slowly
moving particle and this device, would release the spring and the final kinetic energies of the
massive spring and the particle will be larger than their initial one, because the elastic potential
energy was converted into kinetic energy.
The Third Law of Motion blatantly neglects significant inherent nature and characteristics of
body. Thus the Third Law of Motion is true under special or ideal conditions only. Had Newton
thought about these and similar other experiments he would have given the Third Law of Motion
in different way.
4.0 The various types of collisions and the Third Law of Motion.
It is restated that Newton had himself applied to the Third Law of Motion to collisions in third
example. If a body impinges upon another and by its force change the motion of the other, that
body also (because of the quality of, the mutual pressure) will undergo an equal change, in its
own motion, towards the contrary part. However mathematical equations and other requisite
concepts were developed afterwards. Thus now these equations are interpreted. Collisions in
which both momentum and kinetic energy of the system are conserved are called elastic
collisions [3-4]. The coefficient of restitution or coefficient of resilience is unity (e = 1).
Consider two bodies A (projectile) and B (target) of masses M1 and M2 moving along the same
straight line with speeds u1 and u2 respectively. The bodies will collide only if u1 > u2. The final
speeds of bodies A and B are v1 and v2.
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v1 =
M 1  M 2 u1  2M 2u2
M 1  M 2 
(2)
v2 =
M 2 M 1 u2  2M 1u1
(3)
M 1 M 2
If the target is at rest (u2 = 0), then Eqs. (2-3) become
v1 =
M 1  M 2 u1
M 1  M 2 
(4)
v2 =
2M 1u1
M 1 M 2
(5)
The various sub-cases are discussed below.
(i) When M2 >> M1 i.e. target (body B) is very massive compared to the projectile (body A).
The body M2 remains at rest v2=0. Thus
M1 - M2 = -M2,
M2 + M1 = M2
In this case the final speeds of the projectile and target can be calculated from eq.(4) and eq.(5).
v1 =
 M 2u1
= -u1
M2
(6)
Final speed of projectile = -Initial speed of target.
The velocity of target (reaction), v2=0
(7)
The negative sign in v1 indicates that direction of body A reverses after collision. Now action
and reaction are expressed in terms of speed:
Action (u1) = -Reaction (v1)
e=
v2 v1 u1
=
=1
u1  u 2 u1
(8)
As the coefficient of restitution is unity, the collision is elastic. Newton’s Third Law of Motion is
justified. However these are ideal mathematical calculations as we have not considered actual
experimental characteristics at all. The composition of various bodies may be different e.g.
body B can be of cloth and body A of wood, body B can be an air filled ball and body A of
aluminum, body B may be an air filled football and body A of gold, etc. In addition, the point of
impact must be at the centre of the target; if not then the direction of recoil will be different. .
Furthermore, comparative size and point of impact of target and projectile etc. and roughness of
surfaces play significant roles. The various and distinct experiments must be conducted for final
conclusions to be drawn.
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4.1 Mass of body B is 1000 times larger than that of body A
If we consider that the target is 1000 times more massive than the projectile i.e. M2 = 1000M1,
then
v1
(8)
= -0.998001998u1
(final velocity of projectile) or Reaction = - 0.998001998u1 (initial velocity of projectile) or
Action
Action (u1) = -1.002002 Reaction (v1)
e=
(9)
v2 v1
2M 1u1  999M 1u1
=[
]/[u1-0] = 1
u1  u 2
1001M 1
1001M 1
But the action is greater than the reaction. Hence conclusions cannot be drawn from few
observations, and thus other experiments are required to be conducted.
4.2 The rubber ball is dropped on earth
This case is equivalent to elastic collisions (M2 >> M1), as discussed above. Consider a boy
standing, holding a round rubber ball and a cloth ball in different hands. Let the boy is standing
5m above the surface of the earth. Let the boy drops the rubber ball at the earth’s surface with a
force of Faction (1N, say). The ball strikes the wall in time t .
The earth pushes the ball upward towards the boy through a distance 5m in time t. Thus the earth
exerts a force (Freaction = 1N) as reaction. So,
Boy drops the ball through 5m in time t = Ball rebounds from the earth through 5m in time t
Action (Faction = 1N) = -Reaction (Freaction = 1N)
(10)
In this case action and reaction are equal and opposite; hence The Principia’s Third Law of
Motion is justified under these conditions.
4.3 The balls of different compositions dropped on the earth
Let the boy drops the cloth ball (softer, flexible, stretchy or typical) on the earth with a force of
Faction (1N) on the earth, from height of 5m. The ball strikes the earth in time t, and rebounds to
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only distance of 2.5m in the same time t. Thus the earth exerts a force (Freaction = 0.5N) as
reaction. Now compare both the cases, i.e. rubber ball and cloth ball.
Action (Faction = 1N)  -Reaction (Freaction = 0.5N)
In this case, action and reaction are not equal, although opposite. There are
numerous cases like this. Also the earth may be sand, normal clay, or steel or wooden or papers
sheets are placed over it. Each surface offers different reactions.
4.4 Super Elastic Collisions
Consider a specially and typically fabricated round rubber ball and wall (say metallic).
Then a ball is thrown with some force (action) from distance 1m and strikes the wall in 2s (u1=.
0.5m/s). After striking the wall the rubber ball rebounds to initial position at distance 1m in 1s
(v1 = 1m/s). Thus reaction is double than action. This is example of super elastic collisions. It can
be understood in various other way also. For example, nitrocellulose billiard balls can literally
explode at the point of impact. This situation can be realized in different ways. According to
Third Law Of Motion, action and reaction of interacting bodies are always equal i.e. universally
equal, but it is not illustrated in above observations. The above experiments or observations are
discussed in the literature but not in view of Third Law Of Motion. The results from various
experiments are shown in Table I .
The various results are given in Table I.
Table I: Comparison of action and reaction on rubber and cloth balls and elastic collisions.
Sr. No Bodies
Action
Reaction
3rd Law of Motion
1
Rubber ball
F=1N
F=1N
Action = -Reaction
(elastic)
(5m)
(5m)
Cloth ball
F=1N
F=0.5N
(inelastic)
(5m)
(2.5m)
Flour ball or chewing
F=1N
Virtually
gum ball
(1m)
negligible
2
3
(inelastic collision)
Action  -Reaction
Action  Reaction
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Elastic collision
u1
v1
Action(u1) = -
M2 >> M1
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Reaction(v1 )
M2 = 1000M1
u1
v1
Action(u1)
=
 -Reaction (v1)
-0.99800199u1
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Typical plastic ball
u1
v1
=
-2u1
Action(u1)
(super elastic )
 -Reaction (v1)
5.0 Generalized form of the Third Law of Motion
It is a basic principle of science that no conclusions can be drawn on the basis of a single or few
qualitative observations. Results are widely accepted only if they can be replicated, and all
possible values of the parameters are taken in account.
On the basis of the above observations The Principia’s Third Law of Motion is generalized:
“To every action there is always an opposed reaction which may or may not be equal in
magnitude, depending upon the inherent characteristics of the process”;
or
“The mutual actions of two bodies may not always be equal, depending upon the inherent
characteristics of the system, and directed in contrary parts.”
Action  Reaction
or Reaction = - K Action
(11)
K is the coefficient of proportionality and depends upon the inherent characteristics of bodies.
The coefficient of proportionality takes into account the inherent characteristics, nature,
composition, flexibility, elasticity, rigidity, magnitude, plasticity, distinctiveness of interacting
bodies. The bodies may be of steel, wood, rubber, cloth, wool, sponge, spring, etc. The bodies
can be of solid, liquid, gas or mixture of all. The law is applicable for all bodies thus can be
considered for waves as well. Then an attempt may be made to understand the law with help of
eq.(7), as M2>>M1.In Newton’s third law of motion, there is no term which accounts for above
significant factors. It should be mentioned that Newton stated the Second Law of Motion in the
Principia at page 19 in proportionality form (alteration of motion is ever proportional to the
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motive force impressed). Also Newton expressed the Law of Gravitation (F=GmM/r2), and
speed of sound in fluids , Newton’s law of cooling in proportionality form. Finally we deduce
that the generalized form of Third Law of Motion is also in proportionality form. The coefficient
of proportionality, K is consistent with existing coefficients in physics, as discussed below.
Its value can be determined experimentally. The value of the co-efficient of
thermal conductivity Z for different materials is shown below. The quantity of heat Q is
determined by the proportionality method.
Q = ZA [T1-T2]t/d
(12)
The general range of variation of the co-efficient of thermal conductivity Z for
various conductors are:- For Solids: Silver 429 Wm-1K-1, Copper 403 Wm-1K-1, Iron 86
Wm-1K-1, Tin 68.2 Wm-1K-1 Wood 18
Water
Wm-1K-1. For Liquids: Mercury
7.82 Wm-1K-1,
0.599 Wm-1K-1, Acetone 0.167 Wm-1K-1 , C2H5OH 0.226 Wm-1K-1. Likewise the
value of the co-efficient of proportionality K in the Third Law of Motion can be determined.
The universal gravitational constant G is determined experimentally (as it varies so it can be
deemed as constant). Currently the accepted value for the gravitational constant is 6.673 84 ×1011
m3kg-1 s-2, but a recently measured value [5] is much higher, i.e. G = 6.67545 (18)×10-
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m3kg-1s-2. The mass of Earth (M = gR2/G), using the latest value of G, is 0.024% lower.
The determination of coefficient of proportionality is continuous process; the same is true for
coefficient of proportionality K in generalized third law of motion.
6.0
Should action and reaction be directly expressed in terms of physical quantities,
having units and dimensions.
Newton initiated the beginning of some definitions, and laws in systematic and scientific way.
The concept of dimensions was developed by Fourier in 1822 . The unit of force as dyne was
initially defined in 1861 about 184 years after publication of the first edition of the Principia.
The unit of force as Newton was defined in 1948. So Newton could have not explained ‘action’
and ‘reaction’ in terms of force (Newton, dyne) and velocity (cm/s or m/s).
It is obvious that in Newton defined the Third Law of Motion in terms of action and reaction and
expressed in terms of push, pull (force), motion (velocity).
Now the theme of discussion is that should we express action and reaction in terms ,
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of physical quantities which have definite units and dimensions. The possible physical quantities
in this regard may be force, velocity, kinetic energy average velocity, momentum, etc. and the
basic law of conservation of be obeyed in quantitative explanation involving equations. In terms
of average kinetic energy ( as law involves velocity) the may be understood as initial average
kinetic energy is equal to final average kinetic energy ( if there is no inter-conversion of KE to
other energies). Thus whole the topic would be understood afresh. In Newton’s time all kinds of
science (physics, chemistry etc) were lumped together under the name "natural philosophy" and
title of his book is Mathematical Principles of Natural Philosophy. All the experiments need to
be explained quantitatively with well-defined equations and statements like other observations.
7.0 The results of NASA’s Electromagnetic Drive
Rockets are spectacular examples of Isaac Newton’s third law of motion: that to every action
there is an equal and opposite reaction. The forward movement of rocket off the launch pad is
regarded as action, and hot gas out of its engine is reaction. It is assumed (but without actual
measurements) that action is precisely equal to reaction, hence Newton’s third law hold good.
NASA's Eagleworks Laboratory reported significant results for the
EmDrive (the “EM” stands for “electromagnetic”). The EmDrive works without any fuel or
propellants at all; by simply bouncing microwave photons back and forth inside a cone-shaped
closed metal cavity. The EmDrive provides thrust from electricity without consuming
a propellant. This appears to violate well-established laws of physics, such as the third law of
motion; therefore most scientists believed that the EmDrive was impossible [6]. NASA’s
Johnson Space Center reported the thrust data from forward, reverse, and null suggested that the
system was consistently performing at 1.2 ± 0.1 millinewton/kilowatt, which was very close to
the average impulsive performance measured in air [7]. The system is completely reaction less.
The system apparently defies Newton’s third law, which holds that everything must
have an equal and opposite reaction. Generating thrust momentum in one direction requires
something to be expelled, such as a propellant or exhaust, but the EmDrive doesn’t do this.
British aerospace engineer Roger J. Shawyer ( who perceived idea of EmDrive) says
microwaves in EmDrive push on one side more than the other to create thrust. How does it
happen and engine moves forward? If the thrust is produced in both sides is equal then it should
not work. It is not explained.
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Further, it is hypothesized [8] in more fantastic way that EmDrive does produce
exhaust like rockets. It is assumed that the exhaust being blasted out is actually light, or more
specifically, photons that have become paired up with another out-of-phase photon in order to
shoot out of the metal cavity and produce forward thrust. What causes photon to pair up with out
of phase photon? How does it happen? Why do other photons do not shoot out? Further this
explanation is based on assumption that the exhaust photons become invisible from an
electromagnetic point of view, thus are undetectable when come out. The photons can be
detected by other methods as well. It is again unrealistic assumption. The invisible region
extends beyond (380nm and 760 nm ) and microwaves have wavelength 1mm-1m.
If the outer cover of cavity is coated with lamp black or other material, then photons will be
absorbed and drive should not experience thrust, and this hypothesis can be eliminated
easily. Further photons so emitted outwards can be checked with sensitive instruments. It can
serve as test for the hypothesis.
Further third law does not mean the only reaction is necessary for forward
movement, but the action and reaction has to be precisely equal. Hence both must be
precisely measured for final conclusions. Let a satellite weighing equal to 5,000kg is moving
with speed 8,000m/s i.e. momentum 4x107 kg m/s. So photons which pair up with other out
of phase photon and move out of cavity backwards, should have momentum equal to that of
satellite. But in relativity the mass of photon is regarded is zero, otherwise its relativistic
mass (mrel =
M rest
v2
1 2
c
) would be infinity (Mrest >0). Thus, it is not valid hypothesis and
Newton’s third law is contradicted in functioning of EmDrive, hence generalized form can be
used.
The generalized form of the law predicts that reaction could be less than action
[9,10] . In this case reaction is vanishingly small or tending to zero i.e. system is reactionless
practically. In eq.(11 ), the value of K is tending to zero or equal to zero. The action and reaction
can be measured quantitatively for toy aeroplane under controlled conditions, as to measure the
same in space is utmost difficult. For smaller action, more reaction is feasible; if one type of
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energy is converted or transformed to kinetic energy as observed in macroscopic experiments.
The nature, inherent characteristics etc. of interacting bodies are significant but not taken in
account by the third law. Science is dynamic not static.
Acknowledgements
The Author is indebted to various critics; and Professor Robert Donald, Dr Stephen J. Crothers
and Anjana Sharma for suggestions and discussion.
References
[1] I. Newton, Mathematical Principles of Natural Philosophy (printed for Benjamin Motte,
Middle Temple Gate, London, 1727 ), pp.19-20, translated by Andrew Motte from the Latin.
[2]
http://books.google.co.in/books?id=Tm0FAAAAQAAJ&pg=PA1&redir_esc=y#v=onepage&q&
f=false
[3] http://en.wikipedia.org/wiki/Elastic_collision
[4 ] R. Resnick and D. Halliday, Physics Part I (Wiley Eastern limited, New Delhi, 2nd Ed. 1996
reprinted ), pp.215-22
[5] T. Quinn, H. Parks, C. Speake, and R. Davis, Phys. Rev. Lett. 111, 101102 (2013).
[6] https://en.wikipedia.org/wiki/RF_resonant_cavity_thruster
[7] White H et al. On Line Advance publication http://dx.doi.org/10.2514/1.B36120
[8] Grahn P et al. AIP Advances 6, 065205 (2016); http://dx.doi.org/10.1063/1.4953807
[9] Acta Ciencia Indica Vol. XXIV P No.4 127 (1998)
[10] Acta Ciencia Indica Vol. XXV P No.3
113 (1999)