Final answer:
The velocity of a particle as a function of position r after being launched vertically from the surface of Earth is v = v₀√(r₀/r), which indicates that the velocity decreases as the particle moves away from Earth in an inverse square root relationship with r. This explanation is based on the conservation of mechanical energy.
Step-by-step explanation:
To find the velocity of a particle as a function of position r after being fired vertically from Earth's surface, we'll leverage the concept of conservation of mechanical energy.
The particle's initial mechanical energy consists of its kinetic energy, given by ½mv₀² (where v₀ is the initial velocity and m is the mass of the particle), and its potential energy, represented by -GMm/r₀ (where GM represents the gravitational constant times Earth's mass and r₀ is the radius of Earth).
At a distance r from Earth's center, the particle's velocity v can be found by setting its total mechanical energy equal to its initial mechanical energy:
½mv₂ = ½mv₀² - GMm/r₀ + GMm/r,
which, upon solving for v, gives us the expression:
v = v₀√(r₀/r).
This equation tells us that as the particle moves away from Earth (increasing r), its velocity decreases following the inverse square root relationship with r.