Question

The objects listed in the table are dropped from a height of \(85 \mathrm{~cm}\). The height they reach after bouncing has been recorded. Determine the coefficient of restitution for each object. $$ \begin{array}{|l|r|c|l|} \hline \text { Object } & H(\mathrm{~cm}) & h_{1}(\mathrm{~cm}) & \epsilon \\ \hline \text { Range golf ball } & 85.0 & 62.6 & \\ \hline \text { Tennis ball } & 85.0 & 43.1 & \\ \hline \text { Billiard ball } & 85.0 & 54.9 & \\ \hline \text { Handball } & 85.0 & 48.1 & \\ \hline \text { Wooden ball } & 85.0 & 30.9 & \\ \hline \text { Steel ball bearing } & 85.0 & 30.3 & \\ \hline \text { Glass marble } & 85.0 & 36.8 & \\ \hline \text { Ball of rubber bands } & 85.0 & 58.3 & \\ \hline \text { Hollow, hard plastic ball } & 85.0 & 40.2 & \\ \hline \end{array} $$

Short answer

Question: Determine the coefficient of restitution for the given objects dropped from an initial height (H) and reaching height (h1) after bouncing. Answer: Using the formula ε = √(h1/H), the coefficient of restitution (ε) for each object is as follows: 1. Range golf ball: ε = 0.91 2. Tennis ball: ε = 0.71 3. Billiard ball: ε = 0.80 4. Handball: ε = 0.75 5. Wooden ball: ε = 0.60 6. Steel ball bearing: ε = 0.60 7. Glass marble: ε = 0.65 8. Ball of rubber bands: ε = 0.86 9. Hollow, hard plastic ball: ε = 0.68

Step-by-step solution

Write down the formula for the coefficient of restitution

The formula to find the coefficient of restitution (\(\epsilon\)) is: $$ \epsilon = \sqrt{\frac{h_1}{H}} $$ Where \(h_1\) is the height the object reaches after bouncing and \(H\) is the initial height from which the object was dropped.

Calculate the coefficient of restitution for each object

Using the given values of \(h_1\) and \(H\) from the table, we can calculate the coefficient of restitution for each object. Object: Range golf ball $$ \epsilon = \sqrt{\frac{62.6}{85}} = 0.91 $$ Object: Tennis ball $$ \epsilon = \sqrt{\frac{43.1}{85}} = 0.71 $$ Object: Billiard ball $$ \epsilon = \sqrt{\frac{54.9}{85}} = 0.80 $$ Object: Handball $$ \epsilon = \sqrt{\frac{48.1}{85}} = 0.75 $$ Object: Wooden ball $$ \epsilon = \sqrt{\frac{30.9}{85}} = 0.60 $$ Object: Steel ball bearing $$ \epsilon = \sqrt{\frac{30.3}{85}} = 0.60 $$ Object: Glass marble $$ \epsilon = \sqrt{\frac{36.8}{85}} = 0.65 $$ Object: Ball of rubber bands $$ \epsilon = \sqrt{\frac{58.3}{85}} = 0.86 $$ Object: Hollow, hard plastic ball $$ \epsilon = \sqrt{\frac{40.2}{85}} = 0.68 $$ These calculations show the coefficient of restitution for each of the 9 objects.

Key concepts

Understanding Physics Through Coefficient of Restitution

Physics is a foundational science that helps us understand the natural phenomena around us. It involves studying the properties and interactions of matter and energy. A concept that illustrates this interaction well is the coefficient of restitution, denoted as \(\epsilon\). The coefficient of restitution measures how much kinetic energy remains after a collision. In essence, it's a number between 0 and 1 that indicates how elastic a collision is. A coefficient of 1 denotes a perfectly elastic collision, where no kinetic energy is lost, while 0 indicates a perfectly inelastic collision, where objects do not bounce at all after the impact.

To visualize this in a real-world context, consider a basketball dropped from a certain height. The height to which it bounces back is a physical manifestation of the collision's elasticity. In educational practices, experimenting with different materials, such as balls of various types, helps students solidify their understanding of how energy is conserved or lost during impacts.

Decoding Elastic Collisions with Bouncing Balls

Elastic collisions are fascinating events where objects bounce off each other with no loss in the overall kinetic energy of the system. This can be practically observed by dropping balls from a height and watching them rebound. When students calculate the coefficient of restitution for objects, like in the given exercise, they deal directly with elastic collisions. It's crucial to realize that even if there is energy conservation within the system, some energy may still transform into other forms, like sound or heat, leading to values of \(\epsilon\) less than 1.

As an improvement tip, students should always consider the real-world application of concepts like elastic collisions. They can run simple experiments, for example, measuring how different balls bounce on various surfaces, to understand the practical aspect of the energy transformation and restitution coefficient better.

The Principle of Energy Conservation in Collisions

Energy conservation is a key principle in physics, stating that within a closed system, the total energy remains constant over time. This principle underpins the calculations involving the coefficient of restitution. In the context of bouncing balls, the energy conservation principle allows us to predict the height to which a ball will bounce, assuming no external energy loss. Students should recognize that while kinetic energy may seem lost when the balls do not bounce back to their original height, it is merely converted to other forms of energy.

Engaging in solving problems like the provided textbook exercise helps clarify the concepts of energy conservation. However, to enhance understanding, students can be encouraged to apply these principles to various scenarios and closely analyze the factors affecting energy transformation. Discussing real-world applications, such as vehicle crumple zones designed to absorb impact energy in a controlled way, may deepen their appreciation for the energy conservation concept.