The pickleball Coefficient of Restitution (CoR) has become a hot topic in recent weeks, as the USAP has adopted a new Paddle / Ball CoR testing standard (PBCoR). This test was considered necessary in order to reduce the so-called “trampoline effect” in paddles, which was felt to give players an unfair advantage by adding power or “pop” to the ball. This has resulted in the disqualification of some (previously approved) paddles, resulting in recalls and forcing some manufacturers into costly re-designs of their paddles.
Were these disqualifications technically justified? What does the PBCoR test really tell us? Is the PBCoR a true measure of paddle power? To answer these questions, we will look at the science and discuss the technical basis behind the CoR, describe how the CoR is measured, and determine how the CoR relates to paddle performance and power.
What is the CoR?
The Coefficient of Restitution (CoR) is a measure of the elasticity (or “bounciness”) of a collision between two bodies. It is usually determined by the ratio of the relative separation velocity (vs) divided by the relative approach velocity (va) of the two bodies. That is,
CoR = vs / va
In one of the simplest examples, if you drop a pickleball from a height of 78” onto a smooth hard surface (such as polished concrete or stone), by rule it should bounce back at a height of 30-34”. We analyzed this in a previous article and determined that the CoR of a pickleball against a stationary stone surface was about 0.65, corresponding to a fractional energy loss of 58%. That is, 58% of the kinetic energy of the ball is lost in the collision with the stationary surface.
It is important to note that the stationary surface onto which the ball bounces is rigid and has zero velocity before and after collision with the ball. It therefore cannot add or subtract energy or momentum to or from the ball. In other words, since the hard surface does not deform it cannot exhibit a trampoline effect to accelerate the ball upward after contact, nor can it provide a cushioning effect to decelerate the ball upon rebound. The 58% energy loss must therefore be in the ball alone. How does this occur?
All materials exhibit some form of damping (sometimes called “hysteresis”), which is a tendency of materials to dissipate energy under stress and strain. When the pickleball deforms, the polymeric chains that make up the plastic stretch and rub against each other causing friction. If the deformation is large enough, some of these polymeric chains break, resulting in micro-cracks that eventually become large cracks. The internal friction and cracking dissipate energy which is not recovered, resulting in the ball bouncing lower than its initial height on rebound.
The Paddle / Ball CoR
In our previous article, we attempted to quantify the trampoline mode of a pickleball paddle by measuring the rebound height of a pickleball when dropped from a height of 78” onto the face of a paddle that was clamped rigidly to a stone slab. Since the edge guard of the paddle lifted the paddle face off of the surface of the stone by roughly 0.10”, the center of the paddle was free to move in a trampoline fashion to propel the ball upward. Test results indicated that the rebound height of the ball increased by only three inches, corresponding to an increase in rebound energy of only 3%, with a resulting CoR of 0.68.
According to some reports, the USAP is limiting the Paddle / Ball CoR to 0.44, where the rebound velocity of the ball is 26.4 MPH. This comes about through a very simplistic interpretation of the CoR, by multiplying the incoming velocity of 60 MPH by the CoR of 0.44, yielding a return velocity of 26.4 MPH. Readers should immediately see a major inconsistency here. If the CoR of a pickleball against a stationary stone block is 0.65, shouldn’t the CoR of a pickleball against a paddle be greater than 0.65? After all, if the trampoline effect is real, why would a stone block that exhibits no trampoline effect have a greater CoR than that of a pickleball paddle? Let’s look at the math for an explanation.
The PBCoR Test
The USAP PBCoR test uses an apparatus shown in Figure 1.
The USAP PBCoR test is conducted as follows:
- A pickleball paddle is attached to a test apparatus that allows the paddle to rotate freely about a pivot point located 2” from the butt end of its handle.
- Pickleballs are fired at a speed of 60 MPH at the face of the paddle.
- The incoming and rebound speeds of the balls are measured with photodetectors.
Knowing the mass properties of the ball, test apparatus, and paddle, it is possible to calculate the Paddle / Ball CoR based on the measured incoming and outgoing velocities of the balls. We may discuss the mathematical details of this calculation in a future article, but for now, let’s just focus on the fundamental principles of physics.
While we can determine how momentum and energy is exchanged between the ball and paddle in polar coordinates (where the paddle rotates about the pivot point), let’s first simplify the problem by assuming that the paddle and ball undergo translation only in one dimension. This case would be equivalent to the ball contacting the paddle at the paddle center of mass, resulting in a pure translation. This assumption should be valid since momentum is exchanged only during the contact time between the ball and paddle (which we determined is on the order of 4 milli-seconds). Over this short duration, the motion of the ball and the paddle should be practically linear. In our one-dimensional idealization, let’s model the ball as a 1 oz weight fired at a speed of 60 MPH at an 8 oz paddle weight initially at rest on a frictionless surface (Figure 2).
Based on the principle of conservation of momentum, we know that:
mBvBi + mPvPi = mBvBf + mPvPf
Where:
mB = mass of ball (1 oz)
vBi = initial velocity of ball (60 MPH)
mP = mass of paddle (8 oz)
vBi = initial velocity of ball (60 MPH)
vPi = initial velocity of paddle (0 MPH)
vBf = rebound (final) velocity of ball (unknown)
vPf = rebound (final) velocity of paddle (unknown)
Knowing that the CoR is the ratio of the relative separation velocities of the ball and paddle divided by their relative approach velocities, we can substitute this into the momentum equation and calculate the final velocities of the ball and paddle as follows:
vBf = (mB – CoR*mP)* vBi / (mB + mP)
vPf = mB * (1+ CoR) * vBi / (mB + mP)
If the Paddle / Ball CoR is 0.44, then the final velocity of the ball (vBf) is -16.8 MPH, and the final velocity of the paddle (vPf) is +9.6 MPH. But wait a second! As stated above, according to some reports, using the USAP Paddle / Ball CoR of 0.44, the rebound velocity of the ball is -26.4 MPH, not -16.8 MPH. Why is there a discrepancy and where does it come from?
Energy Analysis
The answer lies in the fact that when the ball strikes the paddle, a portion of the energy is lost through damping in the ball, another portion is lost through damping in the paddle, and another portion is lost through the motion of the paddle. The remaining energy is used to rebound the ball off the paddle.
Using the rebound velocity of 26.4 MPH, the above equations find that the true value of the PBCoR must be 0.62, which is more in line with our measurements in our previous article of 0.68, and the findings of other researchers. Using a CoR of 0.62 in the above equations also finds that the rebound velocity of the paddle must be 10.8 MPH. Since our test involved dropping a pickleball from a height of 78”, the velocity of the ball at impact was only 14 MPH. At the higher velocity of 60 MPH, we would expect that the CoR would reduce since the deformation of the ball is greater, causing more internal friction losses in the ball.
Knowing that the ball loses 58% of its energy due to internal damping, we can calculate the energy loss in the paddle alone using the principle of conservation of energy:
½ mBvBi2 + ½ mPvPi2 = ½ mBvBf2 + ½ mPvPf2 + KE(Lost)B + KE(Lost)P
Where KE(Lost)B and KE(Lost)P are the losses due to damping in the ball and paddle respectively. Calculating the energy balance, we find that the following:
- 19.4% of the total energy goes into the rebound velocity of the ball
- 20.5% of the total energy goes into motion of the paddle
- 57.8% of the total energy is lost in damping of the ball
- Only 2.3% of the total energy goes into deformation of the paddle!
These percentages were calculated based on a relatively low initial velocity of the ball (vBi) of about 14 MPH from a drop test. It is likely that the true PBCoR will be reduced at higher ball velocities.
Summary & Conclusions
We examined the Coefficient of Restitution of the pickleball Paddle / Ball interaction and found a large discrepancy in the measured PBCoR based on fundamental physical principles. The 0.44 PBCoR limit defined by USAP is an “effective PBCoR”, based on a simplistic interpretation of the CoR using only the incoming and rebound velocities of the ball. Such a methodology does not account for motion of the paddle, damping of the ball, or deformation of the paddle. When properly accounting for these contributions, we find that the “true PBCoR” is around 0.62, which is in-line with our drop tests and findings of other researchers.
With only 2.3% of the total energy from impact going into the deformation of the paddle, it is unlikely that the implementation of the USAP PBCoR test is a valid measure of paddle power potential. Rather, with 57.8% of the energy being lost in the ball, the PBCoR test is a better measure of balls than paddles. Furthermore, since the ball rebound, ball damping, and paddle motion swamp the paddle deformation in comparison, it will be extremely difficult (if not impossible) to differentiate one paddle from another, much less to disqualify a paddle based on the PBCoR test.
