Newton's Cradle: Explaining 2-Ball vs. 1-Ball Dynamics

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The discussion centers on the dynamics of Newton's Cradle, specifically the behavior of one versus two balls being released. When one ball is released, it collides with the stationary balls, resulting in one ball being ejected on the opposite side. Conversely, when two balls are released, two balls exit the cradle, demonstrating the principles of conservation of momentum and conservation of energy. The mathematical comparison shows that the kinetic energy of two balls moving at speed v is not equivalent to one ball moving at speed 2v, confirming that only two balls can be ejected when two are pulled back.

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when one ball is pulled out, released, and collides with the other four in Newton's Cradle, one ball will pop out the other side. Likewise, when two balls are pulled back, two balls will pop out the other side. Why doesn't one ball pop out, but with twice the speed?

You are to use the concepts of conservation of momentum and conservation of energy to mathematically show that only two balls may be ejected on the other side when two balls are pulled back instead of one ball with twice the speed.i know that both momentum and energy are both conserved because the collisons are elastic, but i just get stuck there

P initial=P finalany help would be appreciated
 
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Compare the energy of two balls with speed v to the energy of one ball with speed 2v. Are they the same?
 

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