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

updated 12 June 2013

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 of an ideal inelastic collision is found to be totally conserved and accounted for during collision. Findings show that the initial kinetic energy of two colliding masses M1 and M2 less the total available energy transferred from the separate frames of velocities V1 and V2 of the two colliding masses into their center-of-mass velocity frame of (VCM) is equal to the final energy of the lumped masses  M1 + M2 in the center-of-mass frame. After the inelastic collision even, the two colliding masses will move jointly at precisely their center-of-mass velocity, a velocity which is unchanging in any closed system of unbound masses. The energy transfer is one that goes from the initial frame of each colliding mass to that of the velocity of the center-of-mass frame of the combined masses.

Keywords: invariance of velocity of the 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 frame 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 (VCM) 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 VCM while the moving mass loses kinetic energy as it decelerates from its initial velocity V to the velocity VCM. 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 VCM

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 VCM 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 VCM of a closed system of masses. 2,4

 Figure 1A: In the above animation, the center-of-mass point moves with unchanging velocity. After the collision, the two masses move collectively as a joint mass unit at the very same velocity VCM of the center-of-mass point.

 Figure 1B: The instantaneous positions of the two colliding mass spheres and their center-of-mass point are indicated at snap shot instants in time, from top to bottom. The linear motion of the center-of-mass point demonstrates the invariance of  VCM.

 Figure 1C: A transfer of kinetic energy occurs at the inelastic impact event. Mass M1 transfers a pulse of kinetic energy and decelerates during which time M2 receives a pulse of kinetic energy and accelerates. All energy transfer events take place according to the law of conservation of energy.

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 VCM. 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 VCM. 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 VCM until they recoil. It is found that, as a consequence of the invariance of VCM 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 VCM. 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 - VCM and  M2 approaches with velocity V2 - VCM

 Figure 1D: A relative frame dependent kinetic energy is illustrated.

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 and the moving wall. 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 VCM

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 VCM 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 VCM = 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 kinetic 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 VCM 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 VCM

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 VCM

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 VCM 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 VCM 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 VCM and the linear motion of the center-of-mass point remains totally unaffected by the collision processes.

On the Effect of Energy Dissipation in the Form of Heat on the Velocity VCM

We shall apply the well known Newton's Law of cooling to examine the heat retention of the stainless steel mass spheres used in this inelastic collision experiment. One of the spheres, each with mass 111.5 grams and diameter 3.750 cm, was selected and deliberately heated to 36°C, a temperature of 14°C above the temperature Tenv of the laboratory environment, recorded to be 22°C during the measurement. The heated sphere was suspended in the lab on 3 sharp pointed pins for thermal isolation and allowed to cool. The temperature Tt was recorded using a digital thermometer that was electrically connected to a miniature thermistor probe. The probe was coupled to the heated sphere using a heat conducting silicon grease. The cooling rate was recorded and summarized in Table 1. The Newton's Law of Cooling, equation (6),

 (6)

states that the cooling rate of a heated mass body is directly proportional to the difference between the temperature of the environment Tenv and the time varying temperature Tt of the cooling mass body.

Stainless Steel Spheres Recorded Heat Dissipation Rate

 time (sec) T(t) (°C) T(t) (°C) Measured Fitted 0.0    60.0  180.0  314.0  360.0  390.0  440.0  480.0  540.0  600.0  750.0  920.0 1163.0 1333.0 1560.0 1860.0 2265.0 2940.0 3300.0 3900.0 4200.0 36.000 36.800 35.000 34.000 33.800 33.400 33.000 32.500 32.400 32.000 31.000 30.000 29.000 28.000 27.000 26.000 25.000 23.500 23.000 22.200 22.000 36.356 35.846 34.873 33.858 33.526 33.314 32.967 32.697 32.301 31.918 31.013 30.070 28.862 28.104 27.194 26.148 24.975 23.502 22.908 22.138 22.032

Table 1. Newton’s Law of Cooling applied to the 111.5 gram mass Stainless Steel Spheres

This experimental result finds that after repeated inelastic collisions between the mass spheres, the measurements could not account for sufficient heat loss to support conventional understanding of the inelastic collision cases. After a careful monitoring of the temperature of the spheres after many collisions, the expected accumulation of heat was not sufficient to support any of the inelastic collision models as published in the literature.

The recorded cooling rate for the stainless steel spheres was carefully done using a simple and easily repeatable experiment. An exponential fit was made to the data using a KaleidaGraph data analysis and graphic program, version 3.52. The mathematical fit to the data is presented in equation (7).

 (7)

The cooling rate was experimentally found to be k=0.00055152°C /sec. The countless number of inelastic collision impacts shows an expected lack of heat buildup in the mass spheres, confirming the correctness of the ideal inelastic collision model presented here. The metal spheres of mass 111.5 grams were slung at the velocity of 666.6 cm/sec, delivering a kinetic energy of 2.47E07 dynes or 2.47 Joules for each shot from spring loaded launcher. There was no noticeable heat or thermal effects on the invariance behavior of the center-of-mass velocity VCM of this ideal inelastic collision experiment. The kinetic energy was conserved and totally accounted for throughout the ideal inelastic collision experiment.

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 VCM.

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 VCM 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 VCM.

Figure 2C: Unchanging Velocity VCM

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 VCM 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 VCM.

Discussion & Conclusions

The invariance of the velocity VCM 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 VCM

• An observer who happens to be in the frame of reference of M2 , as depicted in this example, will note that V2=0 before inelastic collision. At impact the fixed observer will note a Frame Transfer or a kinetic energy exchange between M1 and M2 and an unchanged velocity of the center-of-mass VCM , i.e., an invariance in VCM before and after all inelastic collisions.

• A moving observer of any frame will also note the very same energy exchange between M1 and M2 and that the velocity VCM is invariant before and after the inelastic collision. The lumped masses M1 + M2 will move with velocity VCM .

• The ideal inelastic collision illustrates a hard bump; as depicted in Figures 1B and 1C. The energy exchange between  M1 and M2 occurs such that all energy transfers take place according the law of conservation of energy. Recall that energy can be neither created or destroyed.

• It is important to note that an inelastic collision experiment can be designed such that a soft bump could permit a conversion of the kinetic energy from the mechanical bump into an electrical pulse of energy which can only extract from the available quantity of the Frame Transferable Energy. Even if such a generator were 100% efficient, then the converted energy could never exceed the available quantity of Transferable Energy between the masses M1 and M2 for the ideal inelastic collision case. The velocity of the center-of-mass VCM will, however, remain invariant after a soft bump causing the two masses M1 and M2 to move along curved paths until the frame transfer of energy is totally depleted, still obeying alls laws of the conservation of energy. It that instance, the two masses will move together at velocity VCM in the center-of-mass frame.

• Any Physics that correctly describes all physical processes is totally independent of the observer. (Scientific Method)

The kinetic energy is totally conserved and accounted for in all ideal inelastic collision cases 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