It's 1975, and you've been hired to design a pinball machine for an upcoming event. You are told that it needs to be a one-of-a-kind, flashy pinball machine, fitting for the “pinball wizard” (although you don't really know what that means). The cosmetic design of the pinball machine is pre-determined (a piano-themed pinball machine, for some reason). The layout of the obstacles within the machine has also been planned out. However, you are asked to account for the physical design as well.

Since you are building the machine from scratch, there is a lot to account for to make sure the machine works as it should. There are a few limitations on your design, given here:

1. The pinball must be launched with a spring. 2. The pinball must enter the main platform with a reasonable speed, somewhere between 1 to 2 m/s. 3. The pinball must take at least 1.2 sec to roll from rest from the top of the machine to the flipper bats, based on the length and incline of the table (shorter times than 1.2 sec result in too difficult of a pinball game, even for the wizard).

Your employer asks you to provide a detailed presentation of how the pinball's energy changes over time, preferably in a graph of some kind. Your design must account for an analysis of the following energy types, kinetic energy (translational and rotational) and potential energy (spring and gravitational). Your design should also include some sort of component in which a thermal energy exchange is occurring and have a complete analysis of that exchange. Don't mess this up; the pinball wizard demands perfection!

A good solution is not just equations and numeric predictions but will include commentary and discussion of those equations and predictions and reflections on what those predictions mean. We encourage you to use the simplest model that correctly captures the essential physics.