When a falling meteor is at a distance above the Earth's surface of 2.78 times the Earth's radius, what is its free-fall acceleration caused by the gravitational force exerted on it?
To find the free-fall acceleration of a falling meteor at a distance above the Earth's surface, we can use Newton's law of universal gravitation and Newton's second law of motion.
1. Newton's law of universal gravitation states that the gravitational force between two objects is given by the formula:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force between two objects
- G is the gravitational constant (approximately 6.674 * 10^-11 N*m^2/kg^2)
- m1 and m2 are the masses of the two objects
- r is the distance between the centers of the two objects
2. Newton's second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass:
F = m * a
Where:
- F is the net force acting on the object
- m is the mass of the object
- a is the acceleration of the object
To find the free-fall acceleration of the falling meteor, we need to calculate the gravitational force acting on it by the Earth and divide it by the mass of the meteor.
Given that the distance above the Earth's surface is 2.78 times the Earth's radius, we can calculate the distance between the meteor and the Earth's center. Let's assume the Earth's radius to be 'R'.
Distance from Earth's center to meteor = 2.78 * R
We also know that the mass of the meteor is negligible compared to the mass of the Earth. Therefore, we can ignore the mass of the meteor when calculating the gravitational force, as it will have negligible effect on the Earth's motion.
Now, let's substitute the values into the first equation to calculate the gravitational force between the meteor and the Earth:
F = G * (m_Earth * m_meteor) / r^2
Since the mass of the meteor is negligible, the equation simplifies to:
F = G * m_Earth / r^2
Next, we can substitute the gravitational force 'F' from this equation into Newton's second law of motion equation:
F = m * a
We can rearrange this equation to solve for acceleration 'a':
a = F / m
Substituting the gravitational force 'F' into this equation, we get:
a = (G * m_Earth / r^2) / m
Simplifying further, we get:
a = (G * m_Earth) / (r^2 * m)
Now we have the formula to calculate the free-fall acceleration of the falling meteor, given the distance above the Earth's surface.
The gravitational acceleration varies inversely with the square of the distance from the center of the Earth. If the value at the earth's surface is g, at the higher altitude it is
g/(2.78)^2