Satellites in stable orbits around a massive object, such as the Earth, do not increase their speed with time because they are in a state of dynamic equilibrium between the gravitational force pulling them inward and their tangential velocity that keeps them moving forward.
To understand this, let's consider a satellite in a circular orbit. The gravitational force between the satellite and the Earth provides the centripetal force required to maintain the circular path. According to Newton's law of universal gravitation, the force of gravity is given by:
F_gravity = (G * m * M) / r^2
where G is the gravitational constant, m is the mass of the satellite, M is the mass of the Earth, and r is the distance between the center of the Earth and the satellite.
The centripetal force required for the satellite to move in a circular path is given by:
F_centripetal = (m * v^2) / r
where v is the tangential velocity of the satellite.
For the satellite to remain in a stable orbit, the gravitational force and the centripetal force must be equal:
F_gravity = F_centripetal
(G * m * M) / r^2 = (m * v^2) / r
Canceling out the mass of the satellite (m) from both sides of the equation, we get:
(G * M) / r = v^2
From this equation, we can see that the tangential velocity (v) of the satellite is constant and determined by the radius of the orbit (r) and the mass of the central object (M). The speed does not increase with time because the gravitational force and the centripetal force balance each other, resulting in a stable orbit.
It's important to note that this explanation assumes an ideal scenario without considering other factors like atmospheric drag, perturbations from other celestial bodies, or the effects of general relativity, which may lead to small variations in a satellite's velocity over time. Nonetheless, in the absence of external influences, satellites in stable orbits maintain a consistent speed due to the equilibrium between gravity and their tangential velocity.