In order to impart motion (additional motion) to an object in free space, inertia "soaks up" an given amount of energy prior to additional movement, which is related to its mass, irrespective of any gravitational field. If an existent gravitational field was present when the additional motion is imparted, this amount of energy would also need to be "soaked up" prior to movement. How is this carried (and temporarily stored) energy accounted for in measurement of objects total energy with respect to any gravitational field which may have been present when the force which imparted the motion was first applied? So my question is, when a body travels in free space, how can we ever know it's total energy without knowing from whence it came?
In order to impart motion (additional motion ) to an object in free space, inertia "soaks up" an given amount of energy prior to additional movement, which is related to its mass, irrespective of any gravitational field.
There appears to be some misconception here. If the movement of an object is not obstructed, it starts accelerating as soon as a force is applied to it, i.e., it does not "soak up" some energy before it starts accelerating.
For any given force, heavy objects (big inertia) don't accelerate as much as light objects ($a=\frac F m$), so it may appear that they are not responding to the force right away, but they do, and, as soon as they start accelerating, their speed and, therefore, their kinetic energy changes.
In the presence of the gravitational field, the changing position of the object, due to the gravitational force or other forces, could also result in the change of its potential energy, which is determined by the position of the body relative to Earth or other celestial bodies.
In addition to the translational movement and the associated linear speed and kinetic energy, the object may rotate around its COM with some angular speed and have additional, angular, kinetic energy.
when a body travels in free space, how can we ever know it's total energy without knowing from whence it came?
As discussed, when a body, presumably with a known mass, travels in free space, its total energy (not including its internal energy) consists of its kinetic energy, which is defined by its speed (linear and rotational) and its potential energy, which is defined by its position.
So, the total energy of an object can be calculated without any knowledge of where it came from.
how can we ever know it's total energy without knowing from whence it came?
A consistent version of "Total energy" requires that we have a defined reference for both kinetic and gravitational energy. Neither are defined absolutely, but only in reference to some particular zero point.
Kinetic energy is defined by your observational frame. When the object is at rest in the frame, we declare it to have zero kinetic energy. Another observer in a different frame would measure it differently.
Gravitational energy in the vicinity of the Earth's surface is often measured as height above some reference plane, while in other situations might use the force integral when moving from a point "at infinity" as zero.
As long as these references are done consistently, then the history of an object is no longer needed to calculate the (current) value for total mechanical energy. Instead it means that two objects that began their journey at different points in a gravitational field did not begin with the same energy.