The main equation for you is that of energy conservation:
$$K_1+U_1+W=K_2+U_2$$
The numbers represent before and after, or situation 1 and situation 2.
- Kinetic energy $K$ is "motion" energy. Objects having a speed (not acceration, that doesn't matter) have kinetic energy: $$K=\frac12 mv^2$$
- Potential energy $U$ is "stored" energy. This appears when objects "want to" move somewhere else - in this case the object wants to fall down again because of gravity, so by lifting it up there is "stored" energy which can come in use by letting it go: $$U=mgh$$
- Work $W$ is energy added. And work is done by forces $F$: $$W=F x$$ where $x$ is the distance moved. So, if a force - like the one applied by your hand when lifting the object - lifts the object the distance $x$, then it has done the work $W=Fx$ on the object; in other words, it has added this mount of energy to the system.
To begin with there is no motion and the object is at the bottom, so $K_1=0$ and $U_1=0$. You then do work $W$, which turns into kinetic energy while some of it is also stored as potential energy. The equation becomes:
$$K_1+U_1+W=K_2+U_2
\quad\Leftrightarrow\quad
0+0+W=K_2+U_2
\quad\Leftrightarrow\quad
Fx=\frac12mv^2+mgh$$
So the work is equal to the sum of the final energies in situation two. All that energy comes from the energy input in the form of work.
The acceleration does not have an influence on all this. The only thing acceleration has an influence on is how big the force will be - from Newton's 2nd law, a larger acceleration requires larger forces. So if the object is accelerated largely, then that would have caused a larger force and thus more work $W$ done.
Even just writing this I have thought of many other apparent problems and contradictions which I can't even begin to formulate into a coherent question.
Don't give up!
You can think of gravitational energy being stored in a system of bodies, not just one body or the other. When you apply force over a distance (work) to the ball, it is being stored in the system of "the ball and the Earth." We can capture the concept of this energy stored in the system by saying its "stored in the gravitational field," but at the very minimum we should say that it's stored in the system.
Similar issues show up in electrostatics. In electrostatics, potential energy is almost always between two bodies, not in one or the other. If you choose to think of it as being in one body or the other, you end up in some really peculiar paradoxes.
What makes this tricky to understand intuitively is that we have many cases where one object is so astonishingly massive compared to the other that we can often handwave away this system-wide thinking, and pretend that the ball is the thing that actually has the gravitational potential energy. This is similar to how electrical engineers assume there is such a thing as a "ground" and that it can sink infinite electrical energy (there's a glorious pile of issues like ground loops which are associated with faulty assumptions regarding grounds). However, in many reasonable environments, these simplifications (such as assuming the earth doesn't move in response to us jumping upwards) are effective, so we keep using them.
There are also theories regarding what gravity "is" in general relativity and quantum mechanics. If one wishes, one can pursue those and come to a deeper answer. However, I don't believe they are necessary for everyone to learn.
Best Answer
If you 're pertaining to this sort of example , i think this should be your answer
EG : An object is laid on the ground, it has some potential energy . When I throw it to a higher level , according to the formula of potential energy = mgh ( m-mass, g- acceleration due to gravity , and h- height ) , becuase the height increases , the potential energy increases .
Question is WHY ?
" Energy cannot be destroyed or created it is simply transferred into another form" Ans: When I throw an object to higher level with my hand , my hand gives the object some energy in the form of kinetic energy. As the energy ( kinetic energy) is moving up overcoming the force of gravity, this energy then transfers to the objects potential energy thus increasing the potential energy of the object.
When the object reaches its maximum height , all its left is with potential energy so for a moment only it stays there , and then due to the gravity it accelerates towards the grounds , turning all its potential energy to the kinetic energy .
I hope this clarifies your doubt , If not I suggest you could specify the question for more clarity so that other users can post the right answers for you .