For starters, there's no friction in your equations, when you yourself admit that friction is very important. Hence, the equation cannot be right in the context of TT without the consideration of friction.
Zeio's link to pickleball mechanics actually contains a complete equation and model which is the correct model to use (I actually read it and understood it in full - it's not rocket science, just good old free body diagrams). I bet brokenball doesn't even know how to interpret the article and didn't even read it lol. He only knows how to repeat an equation that primary school kids learn - and the sad thing is he actually thinks he's the coolest guy on the forum for knowing that. I would question his claim of being an engineer, because I work with many, many engineers on a daily basis, and he doesn't have an engineering mindset - period.
A detailed analysis of the role of paddle surface roughness in spin production.
twu.tennis-warehouse.com