Day: August 19, 2019

Bouncy Ball

Physics simulations are highly reccomended to solve this science challenge.

You release a bouncy ball from 1 metre above the Earth. Neglect decrease from surface gravity. It will fall and bounce, then fall again, and so on.With each bounce, the ball’s velocity after the bounce is the negative root of it’s speed before it. For example, if it’s velocity was 4 m/s down, it’s velocity would become -2 m/s down, or 2 m/s up. Clearly, this is not a normal bouncy ball. In fact, it breaks several laws of physics. But still, calculate the velocity of the ball after 100 jumps.

Strange Sums

Algebra and geometry are needed to solve this challenge.

Sigma notation is the topic of today’s challenge. At the bottom, we set a variable, usually ‘K’. K typically starts at 0 or 1. We would write ‘k=0’ or ‘k=1’ below the Sigma. Sigma notation is often used to denote the partial or full sum of an infinite series. At the top, we say when the series will end, often infinity. The value of the expression is the sum of the stuff to the right of the sigma for every value of K. The challenge for today is to find the sum of the expression. Look out for the sequel to this challenge, Peculiar Products, in which we will discuss Pi notation.

\sum_{k=2}^{\infin}\Bigg(\sum_{l=0}^{\infin} \Big(\frac1{k^l}\Big)\Bigg)
Solution