If an object moves with constant absolute and relative velocities, then its acceleration will be zero. This can be mathematically proved using the equations of motion.
Let's assume that the object has an initial velocity (v₀) and a final velocity (v) in the same direction. The absolute velocity is the magnitude of the velocity without any regard to direction, while the relative velocity is the velocity of the object relative to another reference point or object.
The acceleration (a) is defined as the rate of change of velocity over time and is given by the equation:
a = (v - v₀) / t
where t is the time taken for the velocity to change from v₀ to v.
Since the object is moving with constant absolute velocity, the magnitudes of the initial and final velocities are the same:
|v| = |v₀|
Since the object is also moving with constant relative velocity, the relative velocity (v_rel) remains constant. Therefore, the change in velocity (v - v₀) is equal to the change in relative velocity (v_rel).
v - v₀ = v_rel
Substituting this into the acceleration equation, we have:
a = v_rel / t
Since v_rel is constant and not changing over time, the rate of change of v_rel over time (a) is zero:
a = 0
Therefore, if an object moves with constant absolute and relative velocities, its acceleration is zero. This means that the object is not accelerating and its velocity remains constant over time.