Kinetic energy is simply the energy contained in a moving object by virtue of its motion and is given by the equation KE=1/2*m*v^2, where m is the object's mass, and v is its velocity.

The balls shown above are an (approximate) example of what's called an "elastic collision." Elastic collisions are collisions between objects which conserve the total kinetic energy. Obviously these are idealized situations, but the above ball trick is a good approximation to an elastic collision.

There's one more concept you need in order to understand the balls above: the conservation of momentum. Momentum is a quantity contained by a moving object and is given by p=m*v (momentum also carries with it a direction, but we won't worry about that here). In a closed system, momentum is always conserved, just like energy.

In this particular ball system (an elastic collision), the only way to conserve both energy and momentum is for the first ball to transfer all its energy and momentum to the last ball. If one of the middle balls were to start moving, it would have to push a bunch of the other balls with it (since they're all touching), and that would violate one of the two conservation laws.

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To answer your other questions:

is there a limit to the number of balls ?...could it be a hundred ?

There isn't a limit in theory if all the balls are perfectly elastic (no kinetic energy is lost). Energy cannot be created or destroyed in a closed system, so the only way to lose kinetic energy is if its transferred into some other form of energy. In practice, each ball you add probably causes energy to be lost due to friction, so there would be some limit.

The ball then is just a conduit for the energy yes ? and does not have sole ownership of the thing called kinetic energy !

Kinetic energy is an accounting tool: it's like the energy "cash" the ball has by virtue of its speed and mass. If it changes either speed or mass, it will lose or gain some energy.

Say something is stopped very quickly...does the kinetic energy then leave the thing and just continue on outside the thing it was in ?

Kinetic energy only exists in moving objects with mass, so it can't just continue on if an object stops. If something stops quickly, the kinetic energy it had has to go somewhere (since total energy can't be created or destroyed). This could either go into kinetic energy of another object or objects, such as the balls above, or it could be into other forms of energy (such as a car's brakes, which transform kinetic energy into heat* energy through friction).

*Heat is in turn just a measure of the kinetic energy of the atoms and molecules in something as they jiggle about.