Is all motion basically the result of big bang, including atomic and sub-atomic motion?
That depends on what your view on the Big Bang is. If you think it is the creation of everything in the universe, then I guess you could say that the big bang is the "source" of all motion. However your question is ill-posed. It's like asking if the Sun is the source of all flavor since it is the source of energy for all life on Earth, and subsequently the food we eat is the result of life.
I meant that all the four fundamental forces were united previously and then they separated. These basic forces move bodies, sub atomic particles, photons etc. (I am skeptical as to when inertia came into existence, though)
We are also a product of there basic forces. So when we move a thing can we attribute it to the ‘forces’ that made us? I asked the question from this view point.
All forces can be explained in terms of the four fundamental ones. If you looked closely enough at the surface of our palms when we are pushing an object you'll find that all the motion of the object can be attributed to interactions (of the electromagnetic/nuclear sort) occurring between our hand and the object in question.
In that sense, then yes, every 'force' in the universe can be explained by the summation of millions of interactions occurring at the molecular level, all of which are caused by one of the four fundamental forces. That's why we call them fundamental.
any explanation as to why these four fundamental forces took birth? and why only four and why did they get separated, when they were one initially?
There are some theories that claim the four forces were once all "Unified" into one force. This just means that at very high energy and density levels the forces and their force carriers are the same. We don't know of any underlying "why" or how they came into existence. I suspect it has something to do with the creation of the universe in its current state.
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