Both will reach the bottom at the same time.
Explanation
The motion of falling bodies is governed by the force of gravity. In the absence of external resistive forces (such as air resistance), this motion is termed "free fall." The acceleration experienced by a body during free fall is known as acceleration due to gravity ($g$).
Detailed Analysis:
According to the laws of gravitation and motion, the force of gravity ($F$) acting on an object is given by $F = mg$, where $m$ is the mass of the object and $g$ is the acceleration due to gravity. According to Newton's Second Law of Motion, the acceleration ($a$) of an object is the force applied divided by its mass ($a = F/m$).
Substituting the force of gravity into this equation:
$$a = \frac{mg}{m} = g$$
This derivation demonstrates that the acceleration of a falling body is equal to $g$ and is independent of the mass of the body. Consequently, in a vacuum chamber where air resistance is non-existent:
- There is no buoyant force or drag to slow down the lighter object (the paper).
- Both the stone and the paper experience the exact same acceleration ($9.8 \text{ m/s}^2$ near Earth's surface).
- Starting from rest from the same height, they will cover the vertical distance in the same amount of time.
Key Takeaway:
In a vacuum, acceleration due to gravity is constant for all objects regardless of their mass, shape, or size; therefore, all objects dropped simultaneously from the same height will reach the ground at the same time.