6.5. Kinetic Energy with Negative Effective Mass:

In the context of extended classical mechanics, kinetic energy (KE) is intricately tied to the concept of negative effective mass, specifically apparent mass (−Mᵃᵖᵖ) and dark energy's negative effective mass (Mᴅᴇ). This relationship, expressed as KE ∝ −Mᵃᵖᵖ ∝ Mᴅᴇ, indicates that systems exhibiting negative effective mass behaviour lead to unconventional kinetic energy dynamics, deviating from classical interpretations.

Normally, kinetic energy is defined by the relationship KE = 1/2·M·v². In classical mechanics, as velocity (v) increases, a constant mass (M) would lead to an increase in kinetic energy. However, when considering negative effective mass, particularly in systems influenced by  apparent mass or dark energy, this traditional correlation is modified. The repulsive nature of negative effective mass alters the system's energy behaviour: as force increases and the apparent mass (−Mᵃᵖᵖ) grows, the effective mass in the equation F = (Mᴍ + (−Mᵃᵖᵖ))⋅aᵉᶠᶠ results in a decrease in the normal matter mass (Mᴍ) while still allowing for an increase in kinetic energy due to the negative contribution of apparent mass. The total effective mass Mᵉᶠᶠ = Mᴍ + (−Mᵃᵖᵖ) remains positive until Mᴍ = −Mᵃᵖᵖ.

As long as the effective mass Mᵉᶠᶠ remains positive, the system behaves in a manner similar to classical mechanics. However, once Mᴍ < −Mᵃᵖᵖ becomes negative, the kinetic energy undergoes a fundamental shift. In this state, kinetic energy transitions from traditional mechanical behaviour to characteristics akin to dark energy. Consequently, rather than gaining energy with acceleration, the system may exhibit behaviour where kinetic energy appears to decrease or reverse direction, defying classical expectations.

Such counterintuitive effects can manifest as systems gaining kinetic energy while seemingly decelerating or self-propelling against gravitational forces. The interaction between negative effective mass and positive mass in these scenarios leads to unique dynamics where kinetic energy does not operate under the conventional rules dictated by positive mass alone.

These unusual kinetic energy behaviours are primarily observable on intergalactic scales, where dark energy's influence is prominent. In gravitationally bound systems, like galaxies, the effects of negative effective mass are largely overshadowed by the stronger gravitational forces of ordinary and dark matter. Nevertheless, in large-scale cosmic phenomena, the role of negative effective mass becomes crucial for understanding the expansion of the universe and the formation of large-scale structures.

By integrating the concept of negative effective mass into the analysis of kinetic energy, extended classical mechanics offers a deeper comprehension of cosmic dynamics. This perspective challenges conventional principles of energy conservation and paves the way for new experimental and observational insights, particularly in regions dominated by dark energy and dark matter.