A ball is thrown horizontally at a speed of \(\quantity{2}{m\,s^{-1}}\) from a point \(\quantity{1.25}{m}\) above \(O\) which is on a flat horizontal surface. After each bounce it reaches a height which is \(75\%\) of its maximum height after the previous bounce. Ignore air resistance and take the acceleration of gravity, \(g\), to be \(\quantity{10}{m\,s^{-2}}\).

  • Sketch the trajectory of the ball.

  • Calculate \(t_n\), the time the ball is in the air between the \(n^\text{th}\) and \((n+1)^\text{th}\) bounces.

  • How far from \(O\) does the ball get before it stops bouncing?