Significant Findings^©

A Frame Transfer Model shows Conservation of Kinetic Energy for the Ideal Inelastic Collision

updated 15 September 2009

Abstract

A frame transfer model introduced here shows that the kinetic energy is totally conserved and accounted for in the ideal inelastic collision as well as in the elastic collision. The kinetic energy transfer between colliding masses in the ideal inelastic collision case is found to be totally consistent with the law of conservation of energy which states that energy can be neither created nor annihilated. In all inelastic collisions, the two colliding masses move jointly at precisely their center-of-mass velocity, a velocity which is unchanging in a closed system of unbound masses. For this reason, a properly formulated energy transfer model is chosen to be one that goes from the initial frame of the moving mass to that of the center-of-mass frame.

Keywords: invariance of velocity of center-of-mass, conservation of kinetic energy, closed system of masses

Introduction

It has been traditionally taught in the Physics classrooms that the kinetic energy is not conserved in the inelastic collisions.^1-5  This study finds that a frame transfer technique of kinetic energy from a moving mass into the center-of-mass frame of the combine masses at the inelastic collision event shows a conservation of kinetic energy for the inelastic collision case; a hitherto not covered topic in the textbooks. ^2,4,5 

The law of the conservation of energy clearly states that energy can be neither destroyed nor created. In the rest frame, the kinetic energy is by definition that quantity of energy that is deliverable to a receiving mass in that frame. The kinetic energy of a moving mass is a function of its mass and the velocity of its mass referenced to the receiving mass. A moving mass can deliver the total content of its kinetic energy to a resting mass if and only if the moving mass of an initial velocity V were to come to a final velocity of V=0. In order for this to occur, the resting mass must be tied to the rest frame so that it does not move. An occurrence of an ideal inelastic collision between a moving mass and a stationary mass that is free to move will always result in a motion of the joined masses at precisely their center-of-mass velocity (V_CM) of a closed system of unbound masses. This is confirmed by countless numbers of experimental observations. With this, it is clear that the total kinetic energy of a moving mass referenced to the rest frame can not be transferred to the combined masses at the event of an inelastic collision with a mass that is initially at rest and is free to move. The proper frame transfer of kinetic energy must be referenced to the center-of-mass frame, not the rest frame, for the ideal inelastic collision. At collision, the resting mass gains a pulse of kinetic energy as it accelerates from its initial velocity V=0 to the velocity V_CM while the moving mass loses kinetic energy as it decelerates from its initial velocity V to the velocity V_CM. This is an energy exchange bump; a frame transfer of a kinetic energy burst that is in consistency with the law of conservation of energy. The properly applied frame transfer model clearly shows that the kinetic energy is totally conserved and accounted for in the ideal inelastic collision case as well as in the elastic collision case.

On invariance of the Velocity V_CM

The inelastic collision is found to obey the very same laws of the conservation of energy as the elastic collision. It is well known that, in the inelastic case, two colliding masses M₁ and M₂, will move jointly at the velocity V_CM where

  (1)

in any closed system of unbound masses. From countless experiments on the inelastic collisions, two colliding masses M₁ and M₂, move jointly at the velocity V_CM of a closed system of masses. ^2,4 

In Figure 1A (an animation) an inelastic collision process is illustrated, showing the instantaneous positions of two mass spheres and their center-of-mass point at snap shot instants in time. ⁶  The illustration clearly shows the linear motion of the center-of-mass point, demonstrating the invariance of the velocity V_CM. In the time resolve images, the center-of-mass point moves along a linear path and appears to be totally unaffected by any action taking place inside of the closed system of masses. The experimental evidence confirms that after the inelastic collision event, the two masses move collectively as a joint unit at precisely the velocity V_CM. In the elastic collision case, the two colliding masses will move jointly as a single mass unit, for a brief instant in time, precisely at the velocity V_CM until they recoil. It is found that, as a consequence of the invariance of V_CM given by (1) and the conservation of the kinetic energy laws, the following mathematical statement correctly describes the ideal inelastic case:

(2)

On the Transformation of Energy from Frame to Frame

The quantities and are the initial kinetic energies of the masses M₁ and M₂, respectively, as would be noted from the rest frame. These are the quantities of energy that are available to be deposited into the rest frame if the two masses go from their initial velocities, V₁ and V₂, respectively, to a complete stop,  V₁=0 and V₂=0. The quantity is the kinetic energy that the mass M₁ transfers from its frame to the center-of-mass frame as it goes from velocity V₁ to velocity V_CM. This is the quantity of kinetic energy that M₁ transfers to the center-of-mass frame at collision. An observer in the center-of-mass frame would note that the mass M₁ approaches with velocity V₁ - V_CM and  M₂ approaches with velocity V₂ - V_CM. 

In Figure 1D an observer on board the moving wall cart would say that the moving mass approaches with velocity V₁ - V₂. An observer in the rest frame should agree on the same relative velocity and the relative frame dependent kinetic energy between the moving mass of velocity V₁ and the moving wall of unchanging velocity V₂. Thus, all intelligent observers, regardless of the frame of observation, must agree on the Physics of the observation. It is important to note that the observer has nothing at all to do with the Physics of the phenomenon.

The laws of Physics is independent of the reference frame. Similarly, the quantity is the kinetic energy that the mass M₂ transfers to the center-of-mass frame as it goes from velocity V₂ to velocity V_CM. 

The energies and represent the total kinetic energy that is extracted from the initial kinetic energies, and , required to bring the two lumped masses M₁ and M₂ into the center-of-mass frame. From this we can see that for any ideal inelastic collision case, all kinetic energy quantities are totally accounted for. It is very important to note that the experimentally observed invariance of V_CM is a tip off that the kinetic energy in a well designed mass collision experiment has to be conserved in the inelastic collisions as well as in the elastic collisions. The mass collision experiment can now be well designed so as to limit mechanical deformations, energy dissipations due to sparks or thermal shock emissions, acoustical vibrations and sound emissions.

Let us assume that mass M₁ = 1 Kg moving with velocity V₁ = 12 m/s and collides ideally with a resting mass M₂ =2 Kg. From equation (1), we can determine that the velocity V_CM = 4 m/s. We find that the initial kinetic energy of M₁ is 72 Joules (J). This is the available kinetic energy that is transferable to the rest frame. We know that all of this kinetic energy cannot be transferred into to the center-of-mass frame, as the velocity V₁ of M₁ does not go to zero, V₁ = 0, a complete stop requirement for a total dissipation of the available kinetic energy. The initial kinetic energy for the resting mass M₂ is zero. We find that the transferable kinetic energy needed to bring M₁ into the center-of-mass frame is 32 J and the transferable kinetic energy needed to bring M₂ into the center-of-mass frame is 16 J. The remaining energy left from these energy transfers is 72J - 32J - 16J = 24J. The resulting lumped mass kinetic energy from the right hand side of equation (2) is found to be exactly 24J. From this, we can see that the kinetic energy in this inelastic collision is totally accounted for. From equation (2), an expansion, simplification and collection of the energy terms arrives at the kinetic energy equation (3) which leads directly to a valid conservation of momentum equation (4) and (5).

   (3)

(4)

  (5)

This may be considered to be a mathematical proof that the energy is conserved and totally accounted for in all well designed ideal inelastic collision experiments as in the elastic collision experiments.

The following animations illustrate the constancy of the velocity V_CM in a closed system for the ideal (non thermal/acoustic energy dissipation experiment) involving combinations of both elastic and inelastic collisions of high tolerance spheres of equal masses.

Figure 2A: Unchanging Velocity V_CM

The kinetic energy of the moving sphere is propagated through a system of stationary spheres via a combination of elastic and inelastic collisions, while the velocity of the center-of-mass remains unchanged in the closed system. The frame transfer of energy is accomplished by extracting energy conservatively from a moving sphere of an initial frame to the center-of-mass frame of spheres of high tolerance masses that are mechanically adjacent to one another. This is essential for instructions on the subject matter of the frame transfer and the conservation of kinetic energy in both the ideal inelastic and the elastic collision cases.

Figure 2B: Unchanging Velocity V_CM

The kinetic energy of a high velocity mass from left to right and 

the kinetic energy of a low velocity mass from the right to left,

are propagated through 4 stationary mass spheres; 

a conserved kinetic energy exchange.

Figure 2 shows the instantaneous positions of six mass spheres and their center-of-mass point at snap shot instants in time. A combination of both elastic and inelastic collisions of six high tolerance spheres of equal masses is illustrated. As illustrated, a fast moving sphere approaches from the left to the right. A slow moving sphere of color approaches from the right to the left. The kinetic energy of the moving sphere is observed to be propagated through a system of stationary spheres via a combination of elastic and inelastic collisions, while the velocity V_CM of the center-of-mass point remains totally invariant in the closed system of masses. The energy transformation occurs via collision processes. The kinetic energy is passed from the moving mass sphere injecting energy from a moving frame into the to center-of-mass frame. The energy is passed in a conserved manner in a combination of elastic and inelastic collision processes. The energy appears to be propagated acoustically through the 4 adjacent mass spheres of high tolerances and of equal masses, while the apparent linear motion of the center-of-mass point remains totally unaffected. ⁶ 

From the snap shot images of the six spheres, the kinetic energy of a high velocity mass approaching from left to right and the kinetic energy of a low velocity mass of color approaching from the right to left, at impact, seems to transfer kinetic energy by propagating it through the four (4) stationary spheres that are in mechanical contact or touching one another. One sphere passes the energy and the momentum conservatively onto the next sphere via a combination of elastic and inelastic collisions, while V_CM remains strictly invariant. The spheres at opposite ends from one another exchange kinetic energy, totally in a conserving manner, via a series of collisions. One can see the conservation of energy law clearly at work here where it is apparent that one moving mass sphere appears to exchanges its energetic action, passing it across through the 4 stationary spheres to be received by the moving mass sphere at the opposite end; a conserved kinetic energy exchange. Again, the velocity V_CM and the linear motion of the center-of-mass point remains totally unaffected by the collision processes.

On the Effect of Energy Dissipation on the Velocity V_CM

A hypothetical robot trap of sufficiently high tolerance can be designed such that the acoustical vibrations, sound and mechanical dissipations are at a minimum. Let a moving spherical mass of an initial velocity V be trapped by the robot of the same mass. After each bounce, the trapped sphere and the box exchange kinetic energy so that they are at rest and in motion for 50 percent of the time. Note that the velocity of the center-of-mass for the two equal masses is exactly V/2. It is very important to note that if an experimenter were to be placed on board our robot trap and became a part of the closed system of masses, there would be absolutely nothing that the experimenter could do to alter the velocity V_CM.

For a hypothetical case, let us assume that an electrical power generator is designed and placed between the bouncing ball and one of the walls of the trap. Each time the generator is pounded by the bouncing ball, a pulse of mechanical energy is converted to a pulse of electrical energy. Let us assume our experimenter chooses to convert this energy to electromagnetic waves and then transmit this energy out into space. Energy pulses of radio or microwaves would be beamed away from the close system of masses out of a hypothetical window into space. This would serve only to extract mechanical energy from the bouncing ball. The bouncing ball would then gradually slow down, dissipating its kinetic energy. The transmitted pulses of energy would gradually get weaker. This would have absolutely no effect whatsoever on the velocity V_CM of a closed system of masses. The experimenter could choose to design the robot so that the trapped mass sphere would not bounce. The snugly trapped sphere and box would then move together with precisely the velocity V_CM.

Figure 2C: Unchanging Velocity V_CM

Spherical Mass is enclosed by a Robot Trap of the same Mass

A precision robot trap of sufficiently high tolerance can be designed such that the acoustical vibrations, sound and mechanical dissipations are at a minimum. A moving spherical mass of initial velocity V is trapped by the robot trap of the same mass. After each bounce, the sphere and the box exchange kinetic energy so that they are at rest and in motion 50% of the time. Note that the velocity V_CM for the equal masses is exactly V/2. The robot can be designed so that the trapped spherical mass does not bounce. The trapped sphere and box will then move together with precisely the velocity V_CM.

Discussion & Conclusions

The invariance of the velocity V_CM and the law of conservation of energy are found to apply directly to the ideal inelastic collision cases. It has been historically taught that the kinetic energy is not conserved in the inelastic collision case. This teaching has prevailed in the classrooms of Physics for nearly a century now. A destruction or annihilation of energy, still taught in all too many Physics lectures, is contrary to the law of conservation of energy. This study finds that the kinetic energy equations presented in the textbooks and lectures for the inelastic collision case totally omit the energy transfer terms, and ; a clear misrepresentation of the ideal inelastic collision process. In the rest frame, the kinetic energy is by definition that quantity of energy that is deliverable to the rest frame if and only if the moving mass of initial velocity V were to reach the final velocity of V = 0. In the ideal inelastic collision case, it is clearly understood that the final velocity after an ideal inelastic collision is precisely that of the velocity V_CM. With this, it is clearly seen from the frame transfer model presented here that the energy transferred to the combined masses at collision is exactly equal to the combined initial kinetic energy of the colliding masses before collision less the frame transfer of kinetic energy that takes place between the masses of different frames at collision as dictated by the law of conservation of energy. This center-of-mass technique clearly shows that the kinetic energy is totally conserved and accounted for in the ideal inelastic collision case as well as in the elastic collision case.

References

  [1] G. Arfken, Hans Weber, "Mathematical Methods for Physicist". Academic Press, pp. 77-79 (1995)

  [2] R. Serway, J. Faugh, College Physics, Philadelphia - New York, Saunders College Publishing, 5, 157 - 166 (1999)

  [3] John David Jackson, Classical Electrodynamics, 3rd. ed., John Wiley \& Sons, Inc., pp. 27-29 (1998)

  [4] Resnick, Walker, Fundamentals of Physics, 5E, Extended, Wiley (1997)

  [5] Sears, Zemansky, Young and Freedman, University Physics, 10th Ed., Addison-Wesley, (2000)

  [6] E. Dowdye, Published online: http://www.extinctionshift.com/SignificantFindingsInelastic.htm

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