Final answer:
To determine the velocity ratio of two balls falling to Earth from different heights, conservation of energy principles are used. Ball 'a' falls from infinity and ball 'b' from a height of six Earth radii, and their velocity ratio is derived from their kinetic energies on reaching Earth's surface.
Step-by-step explanation:
To find the ratio of velocities for ball 'a' that falls from infinity and ball 'b' that falls from a height of six times the radius of Earth, we can use the conservation of energy principle. For ball 'a', the potential energy at infinity is negligible, and its kinetic energy as it reaches Earth will be the result of conversion from the gravitational potential energy.
For ball 'b', we assume it starts from rest at a height equal to six Earth radii (6R) above the surface, so its initial potential energy is determined by this height. When both balls reach the surface, their potential energy will be minuscule in comparison to their kinetic energy.
The energy conservation equation for ball 'a' is: U_1 = K_a. Since ball 'a' starts at infinity, its initial potential energy (U) can be considered zero, and all of it is converted to kinetic energy (K) at Earth's surface.
For ball 'b', the conservation of energy is: U_2 = K_b, where its potential energy is given by U = mgh (with 'h' as the height from which it falls). The kinetic energy of ball 'b' when it reaches the Earth's surface will be derived from this initial potential energy.
The velocities can be found by taking the square root of the ratio of the kinetic energies because kinetic energy is proportional to the square of velocity.