We have a simple position. Pickleball coefficient of restitution has been justified totally on the grounds of trampoline effect. Trampoline effect has been rationalized as a effect that is a result of paddles stiffness. It's mind-boggling that considering this this test would actually forego the opportunity to measure stiffness. Much to the opposite it considers stiffness to be static. Without considering stiffness the rationale for coefficient of restitution falls apart. The need for it still may be there theoretically. But the grounds for it are totally backwards.
To prove COR is inadequate mathematically, let's combine the equations and show that COR fails to account for paddle properties:
1. **Ball Motion Equation with Paddle Force**:
m₂ dv₂/dt = -F_p = k_p x_p
where:
- m₂ = ball mass
- v₂ = ball velocity
- F_p = paddle force on ball
- k_p = paddle spring constant
- x_p = paddle displacement
2. **Paddle Displacement Equation**:
x_p = v₂ t - ∫(k_p/m₂)x_p dt
Approximate using Taylor series expansion:
x_p ≈ v₂ t - (k_p/m₂)v₁ t²/2
3. **Substitute x_p into Paddle Force Equation**:
F_p = k_p x_p ≈ k_p v₂ t - (k_p²/m₂)v₁ t²/2
4. **Substitute F_p into Ball Motion Equation**:
m₂ dv₂/dt = -k_p v₂ t + (k_p²/m₂)v₁ t²/2
5. **Solve for v₂(t)**:
v₂(t) ≈ v₁ cos(ωt)
where:
ω = √(k_p/m₂) = paddle-ball system angular frequency
6. **COR Equation Fails to Account for ω**:
e = v₂/v₁ = cos(ωt) ≠ √(1 - (ΔK / K₁))
Since COR equation doesn't depend on ω (or k_p and m₂ separately), it fails to capture paddle properties adequately. By making displacement a function of time and velocity we can simplify the equation and directly prove that it is not adequate.
After analyzing a large data set of paddle data we found that indeed if coefficient of restitution is considered to be a function of these variables it can be determined mathematically by (pop^2-power^2). This is expected only if the whole calculation is meaningless and another total proof of the inadequacy of this approach and the necessity of considering stiffness and mass.
The very concept that a test that almost is exclusively composed of ball dynamics could be any indicator of paddle dynamics is flawed in the first place. This test measures the properties of a ball almost exclusively. What the usap has invented says nothing about a paddle that is meaningful or scientific.
The biggest question is why are thousands of dollars of paddles being affected by a a rule that is flawed? That more than anything should be the biggest question on player's minds and truly it's the biggest one on mine.
