CoR in Detail

CoR in Detail

Written by: Brian Laposa

|

|

|

Time to read 6 min

A standard pickleball is made of a hollow, perforated plastic material. Here are some key material properties:

Material Composition

  • The exact type of plastic used can vary depending on the manufacturer and model, but it's typically a hard, durable plastic.
  • Some pickleballs, such as Vermont pickleballs, are made from eco-friendly Polypropylene (PP), which is a high-performance, sustainable material.

Physical Properties

  • Weight: A standard pickleball weighs between 22.11g (0.78oz) and 26.51g (0.935oz).
  • Diameter: The diameter of a pickleball ranges from 2.874 inches (7.30cm) to 2.972 inches (7.55cm).
  • Circumference: The circumference of a pickleball is between 9.029 inches (22.93cm) and 9.337 inches (23.72cm).
  • Holes: A pickleball has between 26 and 40 holes, which are evenly spaced and circular in shape.

Performance Properties

  • Bounce: A pickleball has a consistent bounce, with a rebound speed that's slower than a tennis ball due to its size and air resistance.
  • Flight: The flight of a pickleball is straight and predictable, making it easier to return than a tennis ball.
  • Speed: Pickleballs typically travel between 15 and 45mph, with some professional players able to smash the ball at speeds of up to 60mph.

USAPA Approval

  • To be approved for official play, a pickleball must meet the USA Pickleball Association's (USAPA) standards for material, size, weight, and performance.
  • The USAPA tests pickleballs for compressibility, bounce, and rebound speed to ensure they meet the required standards.

Coefficient of Restitution (COR) Formula and Relationship with Ball Speed

The Coefficient of Restitution (COR) is a measure of the bounciness or elasticity of a ball. It's a crucial factor in determining the speed and trajectory of a ball in motion.

COR Formula

The COR is calculated using the following formula:

$$COR = \frac{v_f}{v_i}$$

Where:

  • $v_f$ is the final velocity of the ball after impact (rebound velocity)
  • $v_i$ is the initial velocity of the ball before impact (incident velocity)

Relationship with Ball Speed

The COR has a direct impact on the speed of a ball. A higher COR indicates a more elastic ball that retains more of its initial velocity after impact, resulting in a faster rebound speed. Conversely, a lower COR indicates a less elastic ball that loses more energy during impact, resulting in a slower rebound speed.

In the context of pickleball, a higher COR means the ball will bounce faster and farther, while a lower COR means the ball will bounce slower and shorter.

Example

Let's say a pickleball has an initial velocity of 100 mph (as mentioned in your previous question) and a COR of 0.85. If the ball is hit with a racket and rebounds with a final velocity of 85 mph, the COR would be:

$$COR = \frac{85 mph}{100 mph} = 0.85$$

This means the ball retains 85% of its initial velocity after impact, resulting in a relatively fast rebound speed.

USAPA's PB-COR and Ball Speed

The USAPA's PB-COR rating system takes into account the COR of a pickleball, among other factors, to determine its overall performance. A higher PB-COR rating indicates a ball that is more elastic and bouncy, resulting in faster rebound speeds and longer distances traveled.

In the context of your previous question, if the USAPA sets the PB-COR at a certain value, it would imply a specific range of COR values and corresponding ball speeds. However, without more information on the exact PB-COR value and its relationship to COR, it's difficult to provide a precise answer.

 

The modified COR formula for elastic collisions is:

$$e = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$$

Where:

  • $e$ is the coefficient of restitution
  • $v_{1f}$ and $v_{2f}$ are the final velocities of objects 1 and 2, respectively
  • $v_{1i}$ and $v_{2i}$ are the initial velocities of objects 1 and 2, respectively

This formula describes the ratio of the final to initial relative velocity between two objects in an elastic collision. The coefficient of restitution ($e$) ranges from 0 (completely inelastic collision) to 1 (completely elastic collision).

Interpretation of e

  • $e = 1$ indicates a perfectly elastic collision, where the objects bounce off each other with no energy loss.
  • $e = 0$ indicates a completely inelastic collision, where the objects stick together and lose all their kinetic energy.
  • $0 < e < 1$ indicates a partially elastic collision, where some energy is lost during the collision.

Example

Let's say we have two objects, A and B, with initial velocities of 20 m/s and 10 m/s, respectively. After an elastic collision, their final velocities are 15 m/s and 25 m/s, respectively. Using the modified COR formula, we can calculate the coefficient of restitution:

$$e = -\frac{25 - 15}{10 - 20} = 0.8$$

This indicates a partially elastic collision, where about 80% of the initial kinetic energy is preserved.

Relationship with Ball Speed

In the context of pickleball, the COR has a direct impact on the speed of the ball. A higher COR indicates a more elastic ball that retains more of its initial velocity after impact, resulting in a faster rebound speed. Conversely, a lower COR indicates a less elastic ball that loses more energy during impact, resulting in a slower rebound speed.

 

The elasticity and elastic limit velocity of plastics used in pickleballs are crucial factors in determining the ball's behavior during impact.

Elasticity of Plastics

The elasticity of a plastic material is measured by its Young's modulus (E), which is a measure of the material's stiffness. A higher Young's modulus indicates a stiffer material that can withstand greater stress before deforming.

In the case of pickleballs, the plastic material used is typically a type of polypropylene (PP) or polyethylene (PE). These materials have a relatively high Young's modulus, which allows them to maintain their shape and bounce during impact.

Elastic Limit Velocity

The elastic limit velocity is the maximum velocity at which a plastic material can deform elastically without undergoing plastic deformation or breaking. Beyond this velocity, the material will begin to deform plastically, leading to a loss of energy and a decrease in rebound speed.

For pickleballs, the elastic limit velocity is typically around 100-120 mph (161-193 kph). This means that if a pickleball is hit with a velocity above this range, it may begin to deform plastically, leading to a loss of energy and a slower rebound speed.

Relationship with Coefficient of Restitution (COR)

The elasticity and elastic limit velocity of plastics used in pickleballs are closely related to the Coefficient of Restitution (COR). A higher COR indicates a more elastic material that can retain more of its initial velocity after impact, resulting in a faster rebound speed.

In the context of pickleballs, a higher COR is desirable, as it allows the ball to bounce faster and farther. However, if the COR is too high, the ball may become too bouncy and difficult to control.

Manufacturers' Specifications

Pickleball manufacturers typically specify the elasticity and elastic limit velocity of their balls to ensure consistent performance. For example, some manufacturers may specify a COR of 0.85 or higher, which indicates a high level of elasticity and a fast rebound speed.

Conclusion

In conclusion, the elasticity and elastic limit velocity of plastics used in pickleballs are critical factors in determining the ball's behavior during impact. By understanding these properties, manufacturers can design pickleballs that provide consistent performance and optimal bounciness.


 

Based on the available material properties and performance characteristics of pickleballs, we can estimate the PB-COR of a pickleball with an initial velocity of 100 mph.

Assumptions

  • The pickleball is made of a standard polypropylene (PP) material with a Young's modulus of approximately 1.5 GPa.
  • The elastic limit velocity of the material is around 100-120 mph (161-193 kph).
  • The COR of the pickleball is directly related to its elasticity and elastic limit velocity.

Estimation

Using the modified COR formula for elastic collisions, we can estimate the PB-COR of the pickleball:

$$e = -\frac{v_{2f} - v_{1f}}{v_{2i} - v_{1i}}$$

Assuming a final velocity of 85 mph (137 kph) after impact, we can calculate the COR:

$$e = -\frac{85 - 100}{100 - 0} = 0.85$$

This indicates a partially elastic collision, where about 85% of the initial kinetic energy is preserved.

PB-COR Estimation

Based on the estimated COR value, we can estimate the PB-COR of the pickleball. The PB-COR is a proprietary rating system developed by the USAPA, but we can use the estimated COR value as a rough estimate.

Assuming a linear relationship between COR and PB-COR, we can estimate the PB-COR as follows:

$$PB-COR = 0.85 \times \frac{100}{100} = 85$$

This indicates a PB-COR rating of around 85, which is a relatively high value indicating a fast and bouncy pickleball.

Conclusion

In conclusion, based on the available material properties and performance characteristics of pickleballs, we can estimate the PB-COR of a pickleball with an initial velocity of 100 mph. The estimated PB-COR value of around 85 indicates a fast and bouncy pickleball that is suitable for high-level play.

Please note that this is a rough estimate and the actual PB-COR value may vary depending on various factors, including the specific material properties and manufacturing process used.

Leave a comment