Final answer:
The modified Newton's second law, where force is proportional to acceleration squared divided by a constant, can explain flat rotation curves, as it diminishes the need for unseen mass (dark matter) to account for constant orbital velocities at increasing distances from the galactic center.
Step-by-step explanation:
The question is asking how a modification of Newton's second law (F = ma) to the form F = ma2/a0, where a0 is a constant, could explain the flat rotation curves of galaxies without invoking dark matter.
n classical physics, the force causing circular motion (centripetal force) is given by F = m*v2/r, where v is the velocity of the orbiting body (e.g., a star in a galaxy), and r is the radius of the orbit. If we assume F = ma and a = v2/r, we can write F = m*v2/r.
However, for a flat rotation curve, the velocity v should stay constant with increasing r, meaning the traditional gravitational force (due to visible matter) decreases with increasing radius which does not align with observations.
The modified law, F = ma2/a0, suggests as acceleration decreases with distance, the force does not decrease as quickly as in the classical inverse-square law, allowing velocity to remain constant without needing additional unseen mass, such as dark matter. Therefore, the modified law can account for flat rotation curves by influencing the dynamics of rotational motion in galaxies.