Newton knew that the force which caused the apple's acceleration (gravity)
must be dependent upon the mass of the apple. And since the force acting
to cause the apple's downward acceleration also causes the earth's upward
acceleration (Newton's third law), that force must also depend upon the mass
of the earth. So for Newton, the force of gravity acting between the earth
and any other object is directly proportional to the mass of the earth, directly
proportional to the mass of the object, and inversely proportional to the
square of the distance which separates the centers of the earth and the object.
We have seen that in the Universal Law of Gravitation the crucial quantity is
mass. In popular language mass and weight are often used to mean the same
thing; in reality they are related but quite different things. What we
commonly call weight is really just the
gravitational force
exerted on an
object of a certain mass. We can illustrate by choosing
the Earth as one of the two masses
in the previous illustration of the Law of Gravitation:
The proportionalities expressed by Newton's universal law of gravitation is
represented graphically by the following illustration. Observe how the force
of gravity is directly proportional to the product of the two masses and inversely
proportional to the square of the distance of separation.
Markov decision processes provide a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control.
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