# Bouncy balls put on top of each other

**Classical Mechanics**Level 4

After the balls are released, and the first ball touches the ground, each ball bounces off its neighboring ball. If \(v_{n}\) is the speed of the \(nth\) ball right after it bounces of its adjacent ball. What is the ratio of the speed of the \(20th\) ball right after it bounces to the \(5th\) ball right after it bounces.

Simply put what is \[\large{\frac { { v }_{ 20 } }{ { v }_{ 5 } }} \]

**Details and assumptions**

All the collisions that take place are elastic collisions.

Assume \({ M }_{ 1 }\gg { M }_{ 2 }\gg { M }_{ 3 }\gg ........{ M }_{ n }\).