Mechanics, Planetary Motion and the Modern Science Revolution 59
If one ball falls on the others at rest momentum is transferred from
ball 1 to 2 to 3 to 4 to 5, with the result that ball number 5 goes flying off
with the same momentum as ball number 1. In second case the
momentum is double because we let two balls fall, which transfers
enough momentum, to allow both balls number four and five to go flying
off with the same final momentum as balls number one and two had
initially.
The conservation of momentum is a consequence of the fact that the
forces between two bodies are equal and opposite. The two concepts are
equivalent, as we have illustrated, by considering bodies, which have
interacted with each other through collision or contact such as the bullet
and the gun, the person and the raft, or the metal balls. Let us now apply
these two concepts to the gravitational interaction where their validity
seems less obvious. Superficial consideration of a rock falling off a
mountain towards the Earth seems to contradict both the principle of
momentum conservation and the idea that the forces between two bodies
are equal and opposite. Before the rock falls there is no momentum but
as a result of gravity the rock falls, develops velocity and hence
momentum. It also appears that the Earth exerts a force on the rock but
what about the equal and opposite force of the rock upon the Earth. The
resolution of the paradox occurs when we recognize that actually the
Earth is attracted to the rock and moves up to meet the rock in the same
way the rock is attracted to the Earth and falls to meet it. Of course we
never observe the Earth’s motion, because the distance the Earth would
move to meet the rock or the speed it would obtain as a result of this
motion would be so small that it could not be detected but it nevertheless
is there. Momentum is indeed conserved but since the mass of the Earth
is approximately 10^25 times greater than the mass of the rock, in order to
conserve momentum its velocity is 10^25 times smaller than the rock and
hence unobservable. Nevertheless each time an object falls to Earth the
Earth is falling up to greet the object in order to conserve momentum.
Another way of analyzing the Earth-falling rock system is to consider
the relation between force, mass and acceleration According to the
principle of inertia a body will move at constant velocity unless a force
acts upon it, which is to say a body’s acceleration is caused by a force.
The more mass a body possesses the more difficult it is to alter its motion
or produce an acceleration. It therefore follows that the force is equal to
the product of the body’s mass times its acceleration. This means that if
two forces of the same strength operate on two bodies with different