The Mathematics of a Ball Bouncing Down a Staircase

The beauty of modeling movement with math

Keith McNulty

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I recently tackled this applied mathematics problem and was quite delighted with the beauty of the answer so I thought I would share my approach to it.

The problem involves a particle being launched off the top of a staircase and bouncing progressively down, hitting each step once. It’s a classic movement we’ve all seen in our day to day lives, so modelling it with math is a fun challenge. Of course, we will use classical mechanics here and we will ignore messy stuff like air resistance and friction, so the answer is a bit idealistic, but still very pretty I think.

The problem

A straight staircase consists of N smooth horizontal stairs, each of height h above the next stair. A particle slides over the top stair at speed U, with velocity perpendicular to the edge of the stair, and then falls down the staircase, bouncing once on every stair. The coefficient of restitution between the particle and each stair is e, where e﹤ 1.

  • Find an expression for the horizontal distance travelled between the n-th and (n+1)-th bounce
  • If N is very large and L is the length of each step of the staircase, find an expression for U.

Drawing a diagram

Initial diagram of the problem

This is an initial diagram I drew of the problem. Drawing a good diagram is always the best way to attack a classical mechanics problem like this.

My diagram makes an important distinction between the horizontal velocity and the vertical velocity of the particle as it progresses down the stairs. Since gravity and restitution (bounce) are the only external forces at work in this problem, and since they both act vertically, we can conclude that the horizontal velocity of the particle is always U. Vertical velocities at the point of the bounce on each step are notated as v_n.

Therefore the key to solving this problem is to:

  1. Understand the time t_n taken for the particle to move from the n-th bounce to the (n+1)-th bounce by studying its vertical movement.
  2. Using this to determine the horizontal distance d_n between the n-th bounce and the (n+1)-th…

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Keith McNulty

Pure and Applied Mathematician. LinkedIn Top Voice in Tech. Expert and Author in Data Science and Statistics. Find me on LinkedIn, Twitter or keithmcnulty.org