The relationship between an object's mass and the gravitational force and escape velocity on Earth can be explained by the laws of gravity and Newton's laws of motion.
- Gravitational Force: 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 represented as:
F = (G * m1 * m2) / r^2
Where: F is the gravitational force between the two objects, G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2), m1 and m2 are the masses of the two objects, and r is the distance between the centers of the two objects.
On Earth's surface, the gravitational force acting on an object near the surface can be simplified as:
F = (G * M * m) / R^2
Where: F is the gravitational force, M is the mass of the Earth (approximately 5.972 × 10^24 kg), m is the mass of the object, and R is the radius of the Earth (approximately 6,371 km).
- Escape Velocity: Escape velocity is the minimum velocity an object needs to escape the gravitational pull of a celestial body. On Earth, the escape velocity depends on the mass of the Earth and the distance from the center of the Earth to the object's location. Mathematically, it can be calculated using the following formula:
Ve = √(2 * G * M / r)
Where: Ve is the escape velocity, G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the object's location.
In summary, an object's mass affects both the gravitational force and escape velocity. A greater mass of an object leads to a stronger gravitational force between the object and the Earth. Similarly, a higher mass of an object would require a higher escape velocity to overcome the Earth's gravitational pull and escape into space.