Infinite series in context: bouncing balls and nested squares Problem set A

A ball is dropped from a window 15 m above the ground. After each bounce it rises to 60% of the height of the bounce before. Find the height it reaches after the 8th bounce.

A ping-pong ball is dropped onto a table and takes 1.5 s to fall. Each time it bounces it takes 80% of the time of the previous fall to rise, and the same time to fall back.

1. (a) Show that the total time of the motion is given by \(1.5+3(0.8)+3(0.8)^2+\dots\) seconds.
2. (b) Find a formula for \(S_n\), the total time of the first n terms.

A ping-pong ball is dropped onto a table and takes 1.5 s to fall. Each time it bounces it takes 80% of the time of the previous fall to rise, and the same time to fall back. Find the time from the drop until the ball comes to rest.

Given that \(x+\frac{x^2}{5}+\frac{x^3}{25}+\dots=4\), find the value of x.

A tennis ball is dropped from a balcony 20 m high. After each bounce it rises to 70% of the height of the bounce before. Find the total distance it travels before it comes to rest.

A square has sides of 10 cm. The midpoints of its sides are joined to make a new square, and the four corner triangles that are left over are shaded. The same is done in the new square, and again, as shown. Find the total shaded area after 6 stages.

A square has sides of 10 cm. The midpoints of its sides are joined to make a new square, and the four corner triangles that are left over are shaded. The same is done for ever. Find the total shaded area.

A ball is dropped and bounces. The total time of the first n terms of its motion is \(S_n=13.5-12(0.8)^{n-1}\) seconds. Find the least value of n for which \(S_n>12\).