In a vacuum, both heavy and light objects fall at the same rate because the only factor affecting their motion is gravity. The force of gravity acting on an object is determined by its mass, and in a vacuum, all objects experience the same gravitational force regardless of their mass.

$$ F = Gm_1m_2/r^2 $$

where F is the gravitational force, G is the gravitational constant (6.674×10^-11 N m² kg^-2), m1 and m2 are the masses of the two interacting objects, and r is the distance between the objects' centers of mass.

When an object is in a gravitational field, the only force acting on it is the force of gravity. In the absence of air resistance or other external forces, all objects will accelerate at the same rate in the same gravitational field, regardless of their mass. This acceleration due to gravity is denoted by g and is approximately 9.81 m/s² on Earth.

So, in a vacuum, a heavy object and a light object will both experience the same gravitational force and accelerate at the same rate of 9.81 m/s² toward the Earth's center. This is why they appear to fall at the same rate.