Why is acceleration the same for all free-falling objects regardless of mass and distance? Please don't say according to this formula or something. could you please actually explain the theroy behind it.
It is not the same regardless of distance. As one moves away from the surface of the earth, the acceleration is less.
The idea here is in Potential Theory, and I remember Kelloggs massive Dissertation on it.http://books.google.com/books?id=TxlfQi46CvEC&dq=kelloggs+potential+theory&printsec=frontcover&source=bl&ots=sxjfNE5pOX&sig=rMCYmWeiEwczNzjSL3P3oXfk6vo&hl=en&ei=E8_USaDYO6LwswO6k9GkCg&sa=X&oi=book_result&ct=result&resnum=1#PPP9,M1
And Newton wrote on his Potential Theory also, but the writing is not easily clear.
The Theory of Potentials lays out the potential field for matter, and in gravity, one gets the potential function as a function of 1/distance. So when one leaves Earth, the potential of Gravity decays as the reciprocal of distance from the center of Earth. Thus, when one is two Earth radii from Earth, the potential has decreased by 1/2
Now it is this potential which causes the gravitational field to vary, and of course the gravitational field causes forces, which induce motion (acceleration). The potential function is equivalent to stored energy in space, and it is equal over small increments in space as force*changesindistance.
given this, then we have a known (ignoring the calculus involved)
force is proportional to 1/distance^2
and thus, acceleration is proportional to force (Newtons second law).
Now if one goes great distances from Earth, for instance in high altitude orbits (2000km), the difference in acceleration due to gravity is easily measured. But changes on the surface of Earth itself (sea level vs an altitude of 300 m) is not easily deteceted, as the change in distance from the EArth center is trival.
So if I dropped the same ball twice at different heights suppose 1m and 500m the ball that was dropped at 500m would have a lower acceleratino but greater velocity?
2. A square of side is removed from one corner of a square sandwich that has sides of length L. The center of mass of the remainder of the sandwich moves from C to C’. The displacement of the x coordinate of the center of mass (from C to C’) is
1/12 * L
Certainly! The theory behind why acceleration is the same for all free-falling objects regardless of mass and distance can be explained using the concept of gravitational force and the Principle of Equivalence.
1. Gravitational force: When an object falls freely near the Earth's surface, it experiences a force known as gravitational force. This force is responsible for the object's acceleration. Gravitational force depends on two factors: the mass of the object (m) and the gravitational field strength (g).
2. Principle of Equivalence: The Principle of Equivalence, proposed by Albert Einstein as part of his theory of general relativity, states that gravitational and inertial mass are equivalent. In simpler terms, it means that the effects of gravity are indistinguishable from the effects of being in an accelerated frame of reference.
Based on these principles, we can explain why acceleration is the same for all free-falling objects:
1. Independence of mass: The gravitational force acting on an object is directly proportional to its mass. However, according to the Principle of Equivalence, a more massive object has more inertia (resistance to change in motion). As a result, the greater mass cancels out the greater force, resulting in the same acceleration for objects of different masses.
2. Independence of distance: The gravitational force also depends on the distance between objects. However, when an object falls towards the Earth's surface, the distance traveled is relatively small compared to the Earth's radius. Consequently, any differences in gravitational force due to varying distances are negligible. Therefore, all free-falling objects experience the same acceleration regardless of their starting position.
In summary, the theory behind why acceleration is the same for all free-falling objects, regardless of mass and distance, relies on the concepts of gravitational force and the Principle of Equivalence. This combination explains why the effects of gravity result in all objects experiencing the same acceleration during free fall.