CoR in Detail
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Time to read 6 min
A standard pickleball is made of a hollow, perforated plastic material. Here are some key material properties:
Material Composition
Physical Properties
Performance Properties
USAPA Approval
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:
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:
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
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
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.