The new instantaneous velocity of the ball becomes the negative square root. This bouncy ball defies physics, but we can still calculate it’s velocity after 100 jumps. Most importantly, The ball’s velocity at the end of a bounce is -1 times it’s velocity at the beginning of it. So the velocity is square-rooted every bounce. If a non bouncy object is dropped from a height of A metres, and it’s terminal velocity is B metres per second, then B = √(2gA)/2. g is the force of gravity, 9.807 metres per second squared. The initial height, A, is 1 metre. 2gA is 19.614 m2/s2. Thus B is 4.429 m/s. Square-rooting 99 times is the equivalent of raising it to the power of 2-99. We get approximately 1 metre per second upwards. It’s slightly more, but very little.
Day: August 25, 2019
Solution to Strange Sums
Let’s solve an equation! The sum of a decreasing geometric series is described by the second equation. If you would like to know why, see Peculiar Pets. Knowing this, the full equation is clearly the harmonic series, 1/1 + 1/2 + 1/3 + … . The harmonic series adds to infinity, our answer.
\sum_{k=2}^{\infin}\Bigg(\sum_{l=0}^{\infin} \Big(\frac1{k^l}\Big)\Bigg)\\[8pt] \sum_{l=0}^{\infin}\Big(\frac1{k^l}\Big)= \frac1{k-1}\\[8pt] \sum_{k=1}^{\infin}\Big(\frac1{k}\Big)= \infin\\[8pt]