Introduction:
When a satellite is projected vertically upwards from an earth station, the force acting on the satellite due to the earth gradually decreases as it moves away from the earth's surface. In this explanation, we will determine the height above the earth's surface at which the force on the satellite is reduced to half its value at the earth station.

Key Points:
To solve this problem, we will consider the following key points:

1. Gravitational Force: The force acting on the satellite is the gravitational force between the satellite and the earth. This force is given by Newton's law of universal gravitation: F = G * (m1 * m2) / r^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses of the two objects (satellite and earth, respectively), and r is the distance between the objects.

2. Inverse Square Law: According to the law of universal gravitation, the force between two objects decreases with the square of the distance between them. This means that as the satellite moves away from the earth's surface, the force acting on it decreases.

3. Half the Force: We need to determine the height at which the force on the satellite is reduced to half its value at the earth station. Mathematically, this can be expressed as F/2 = G * (m1 * m2) / (r^2), where F/2 is half the force at the earth station.

Calculation:
To find the height at which the force is reduced to half, we can equate the two expressions for force:

G * (m1 * m2) / (r^2) = G * (m1 * m2) / (2r)^2

Simplifying the equation, we get:

1 / r^2 = 1 / (2r)^2

Cross-multiplying and simplifying further:

(2r)^2 = r^2

4r^2 = r^2

3r^2 = 0

This equation has no real solutions, which means that there is no height at which the force on the satellite is reduced to half its value at the earth station.

Conclusion:
Based on the calculations, we can conclude that there is no specific height above the earth's surface at which the force on the satellite is reduced to half its value at the earth station. The force decreases continuously as the satellite moves away from the earth, following the inverse square law.