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If an impact is an event spanning an infinitesimally small time interval, is the total kinetic energy of two colliding objects conserved through an impact? What about the kinetic energy of each individual object?

Step-by-step

Answer to the first question. In general, the kinetic energy of a two colliding particles is not conserved during the impact. To explain why this is the case, let’s begin with observing that we model impacts as events that take place in an infinitesimal time interval and that cause the colliding objects to change velocity but not position. This implies that there cannot be a change in potential energy through an impact for any of the colliding objects. With this in mind, applying the work energy principle, the difference between the total pre- and postimpact kinetic energies of the system measures the work done during the impact by the impulsive forces acting on the system. The total kinetic energy would be conserved if the total work done by the impulsive forces were equal to zero. In turn this would happen if the work done during the deformation phase of the impact were equal and opposite to the work done during the restitution phase. Such a case corresponds to the case in which the collision is perfectly elastic, i.e., the COR e = 1. Hence, whenever the impact is not perfectly elastic, the total kinetic energy of the system is not conserved.

Answer to the second question. In general, the kinetic energy of the individual particles is not conserved during the impact. The explanation for this answer can be given by consider an example in which a moving particle collides with a stationary particle. After the collision, the particle that was initially stationary would be moving, and this clearly indicates that the kinetic energy of that particle has changed. In general, such an impact would cause a change in the speed of the particle that was initially moving so that the kinetic energy of his particle also changes from before to after the impact.

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