Gravity and force I have a question and am not able to answer it.


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*Suppose you drop two objects from different heights. They are exactly the same shape, size and weight. Now we know that the gravitational acceleration is constant for all objects. Also, $F = ma$. Since both the values are the same, they ought to exert the same force on Earth on striking which is not true. You may imagine the difference between someone dropping a ball on you from a feet and one from 10 feet. Can you explain the reason?

*And what exactly is force? Is it the strength for the smallest time or for a second?
If we apply Newton's second law to my question, then the ball dropped from a height should remain in contact in the ground for a very short time and bounce off quickly?
 A: The balls, when dropped, have their own inertia, which is resisting the force of gravity. When the balls are released, they accelerate at a rate of 9.81 m/s squared, towards the maximum speed of terminal velocity - but they would need to be dropped from nearly 2,000 feet up to reach this speed.
In summary, the ball falling from one foot will reach a lower final speed relative to the ground than the ball falling from ten feet up. The phrase relative to the ground is something to keep in mind here. If the ground were not able to resist the downwards force of gravity with the help of the other forces, there would be no impact.
A: 
Since both the values are the same, they ought to exert the same force on Earth on striking which is not true. 

Actually, the values are not the same. The gravitational acceleration is equal for them both, but the acceleration in $\sum F=ma$ is not this gravitational acceleration.
When something hits the ground, it is stopped. That is, it is slowed down very fast from it's impact speed to no speed. That is, it experiences a very large deceleration. And it is this acceleration, you plug into the $\sum F=ma$ formula.
Now, surely, the object with higher impact speed, which is the object that falls from the largest height, is decelerated more and therefore experiences a much larger force on impact, to cause this deceleration.

If you are familiar with momentum $p$, this is a better way to explain this. Because Newton's 2nd law can be written in terms of momentum $p=mv$:
$$\sum F=ma=m\frac{\Delta v}{\Delta t}=\frac{\Delta (mv)}{\Delta t}=\frac{\Delta p}{\Delta t}$$
Now the formulas says that you have a larger force if you have a larger change in momentum (which depends on the impact speed, and therefore on the start-height) or if you have a smaller change in the impact/collision time $t$ (which means that landing on a pillow, where the collision and the slowing-down is stretched over a longer time span than when landing on hard ground, will cause less force).
(The change in momentum is called impulse, which is why you hear this term often.)

And what exactly is force? Is it the strength for the smallest time or for a second?

The simple explanation: Force is what causes motion change or Force is what causes acceleration.
Don't think about it as strength, because many other parameters than strength of a material can cause force - if a material is not very strong but rather very hard, this also cause great force on impact. And if you are flying in freefall, the force of gravity pulls you down, and I don't think you can use a "strength" analogy here. You could maybe rather say that strength of materials is only one subsection of the topic of forces.
A: You can imagine the elastic collision of each of the two objects when they hit the ground. The kinetic energy of each of the two objects makes the difference. the two objects with the same mass have the same force when they hit the ground but not the speed and energy. 
A: What is the force F=mg exerted on the ball?  It is the approximation of M1m2G/r2, of the force of the total mass of the earth on a ball supposed suspended at  height r. The potential as the ball falls changes as M1m2/r,  the potential energy becomes kinetic energy and momentum.
Force instanteneously on a moving object  is dp/dt, the impulse.
The forces on the ball on touching the ground itself is still mg, as it is continuously as the ball is falling, but because it has kinetic energy the impulse dp/dt has to be added to the force exerted by gravity on contact. It is not mg that is sending the ball bouncing back elastically, but conservation of energy and momentum.
If the balls fall from different heights, then the impulse which sends them bouncing back is different, the one falling from a larger height having a greater impulse.
