A Frame Transfer Model shows Conservation of Kinetic Energy for the Ideal Inelastic Collision
updated 12 June 2013
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.
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.
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
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
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:
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.
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.
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.
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.
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),
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.
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).
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
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 kinetic energy is totally conserved and accounted for in all ideal inelastic collision cases as well as in the elastic collision case.