Say you push a mass with 10 N force (just once and let it go) of 10kg object in space with zero friction and nothing in the way of its path. What would be the velocity of that object?
It depends on what you mean by “just once and let go” and the reference frame in which the velocity is calculated. In other words, what was the duration of the force (t) or what was the distance (d) over which the force was applied and with respect to what point in space the measurement is made.
Let’s say the frame of reference is a point in space at the same location as the mass before the force was applied to the mass. We can calculate the velocity in a number of ways, one is to use basic kinematics (given the duration of the application of the force) and the other using the work energy principle (given the distance through which the force acts)
Let’s say the force is applied for 1 second, then for $F=10N$ and $m=10kg$, then
The work-energy principle states that the change in kinetic energy of an object is equal to the net work done on the object.
Since the initial velocity is zero (with respect to the frame of reference chosen) we have
Let’s say the force is applied over a distance of 0.5 meter.
I would assume that if you are putting a "constant" force of 10N, then yes theoretically the object should continue accelerating forever right?
No, because the maximum possible speed is the speed of light. But that’s the subject of special relativity.
But what if you just give an initial push of 10N to the object and let it go so that it floats in space how would you calculate its definite velocity? I am assuming the time in which the object is in the contact with a pushing force will matter is that correct?
I think my answer to the first question answers this as well
Hope this helps.