The speed of an object falling toward Earth due to gravitational acceleration depends on several factors. However, the key factor influencing the speed is the distance between the object and the Earth's center.
According to Newton's law of universal gravitation, the gravitational force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, it can be expressed as:
F = (G * m1 * m2) / r^2
Where: F is the gravitational force between the two objects, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
When an object is falling towards the Earth, it is subject to the gravitational force exerted by the Earth. As the object falls closer to the Earth's surface, the distance (r) between the object's center and the Earth's center decreases.
Now, if we consider a simplified scenario where the object is falling from a significant height close to the Earth's surface (so the change in distance is relatively small), we can approximate the speed using the concept of free fall.
In free fall near the Earth's surface, neglecting air resistance, all objects experience the same acceleration due to gravity, denoted by 'g.' On the surface of the Earth, the average value of gravitational acceleration is approximately 9.8 meters per second squared (m/s^2).
As an object falls freely, its speed increases by approximately 9.8 m/s for each second of fall. This acceleration remains nearly constant (although it slightly decreases with increasing altitude due to changes in the Earth's gravitational field). Hence, the speed of the falling object increases with time.
However, in a more general sense, if we consider objects falling from different distances, the speed will vary based on the distance from the Earth's center. As an object falls closer to the Earth's surface, the distance decreases, resulting in a stronger gravitational force. Consequently, the object will experience a higher acceleration and, therefore, a greater increase in speed.
To summarize, the speed of an object falling towards Earth due to gravitational acceleration does depend on the distance from the Earth's center. The closer an object is to the Earth's surface, the faster it will fall due to the increasing gravitational force.