The gravitational force of the Earth acting on an object with more mass is stronger than the force on a smaller mass object of the same size, yet they both fall at the same rate of increasing speed (i.e. balls of different masses but with similar sizes fall to the ground at the same time).

Why?

The reason why objects of different masses fall at the same rate of increasing speed is due to a fundamental principle in physics called the "Equivalence Principle" and the laws of motion.

Let's break down the explanation step by step:

1. Gravitational Force: Gravity is the force of attraction between two objects with mass. The greater the mass of an object, the greater the gravitational force it experiences. So, in this case, the Earth exerts a stronger gravitational force on an object with more mass compared to an object with less mass.

2. Acceleration due to Gravity: Now, when an object falls, it accelerates due to the force of gravity. This acceleration is denoted by "g" and is approximately 9.8 meters per second squared (m/s²) on the surface of the Earth. Regardless of their mass, all objects near the Earth's surface experience the same acceleration due to gravity.

3. Newton's Second Law of Motion: According to Newton's second law, the force acting on an object is equal to its mass multiplied by its acceleration (F = m * a). In this case, let's consider the force of gravity as the force acting on the falling object. So, the force of gravity (weight) on an object with more mass is greater than the force on an object with less mass, but the acceleration due to gravity is the same for both objects.

4. Canceling of Mass: Now, if we rearrange the equation from Newton's second law (F = m * a), we can find that acceleration (a) is equal to force (F) divided by mass (m). Since the gravitational force (F) is greater for the object with more mass, its acceleration (a) is larger than the object with less mass. However, the mass (m) of the object is also larger, and they cancel each other out.

5. Canceling of Mass again (commonly referred to as "mass drops out"): In equations where the mass of an object is divided by the mass itself, it cancels out. So, when we substitute the force of gravity divided by the mass into the equation for acceleration, the mass term drops out. This means that the acceleration due to gravity is independent of the mass of the object, resulting in both objects falling at the same rate.

Therefore, although the force of gravity is stronger on the object with more mass, the fact that the acceleration due to gravity is the same for both objects leads to both objects falling at the same rate of increasing speed. This principle was famously demonstrated by Galileo Galilei with his theoretical experiment involving two differently weighted balls dropped from the Leaning Tower of Pisa.