91Ó°ÊÓ

The coefficient of restitution \(\epsilon\) for a head-on collision is defined as the ratio of the relative speed after the collision to the relative speed before. (a) What is e for a completely inelastic collision? (b) What is \(\epsilon\) for an elastic collision? (c) A ball is dropped from a height \(h\) onto a stationary surface and rebounds back to a height \(H_{1}\) . Show that \(e=\sqrt{H_{1}} / h .\) (d) A properly inflated basket-ball should have a coefficient of restitution of \(0.85 .\) When dropped from a height of 1.2 \(\mathrm{m}\) above a solid wood floor, to what height should a properly inflated basketball bounce? (e) The height of the first bounce is \(H_{1}\) . If \(\epsilon\) is constant, show that the height of the \(n\) th bounce is \(H_{n}=\epsilon^{2 n} h .(f)\) If \(\epsilon\) is constant, what is the height of the eighth bounce of a properly inflated basketball dropped from 1.2 \(\mathrm{m} ?\)

Step by step solution

01

Understanding Restitution in Inelastic Collisions

For a completely inelastic collision, the two objects stick together after the collision, thus moving with the same velocity. This results in a relative speed of zero after the collision while there was a relative speed before the collision, hence the coefficient of restitution, \( \epsilon \), equals 0.

02

Coefficient of Restitution in Elastic Collisions

In a perfectly elastic collision, the relative speed after the collision is equal to the relative speed before the collision. Therefore, the coefficient of restitution, \( \epsilon \), equals 1.

03

Relating Bounce Height and Restitution for Free Fall

When a ball is dropped from a height \( h \) and rebounds to a height \( H_1 \), energy considerations show that \( \epsilon = \sqrt{ \frac{H_1}{h} } \). This results from equating the kinetic energy during impact to the gravitational potential energy: \( H_1 = \epsilon^2 h \).

04

Calculating Rebound Height with Given Restitution

Given \( \epsilon = 0.85 \) and \( h = 1.2 \; \text{m} \), the rebound height \( H_1 \) is calculated using the formula from step 3: \( H_1 = \epsilon^2 h = (0.85)^2 \times 1.2 \; \text{m} \approx 0.867 \; \text{m} \).

05

Generalizing Bounce Height for Constant \( \epsilon \)

If \( \epsilon \) is constant, the height of the nth bounce \( H_n \) can be generalized to \( H_n = \epsilon^{2n} h \). This progression follows from repeatedly applying the restitution formula across multiple bounces.

06

Computing the Eighth Bounce Height

Using the formula for the nth bounce, for \( n = 8 \) and \( \epsilon = 0.85 \), the eighth bounce height is \( H_8 = (0.85)^{16} \times 1.2 \approx 0.094 \; \text{m} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inelastic Collision

An inelastic collision is an event wherein colliding objects do not bounce away separately from each other after impact. Instead, they stick together and move as a single unit post-collision. This peculiar behavior occurs primarily because the kinetic energy of the colliding bodies is not conserved. Rather, it gets converted to other forms, like heat or internal energy.

The coefficient of restitution, represented by \( \epsilon \), is a parameter that helps quantify the elasticity of a collision. For a perfectly inelastic collision, \( \epsilon \) is equal to 0. This indicates that the relative velocity between the objects post-collision is zero, as they combine and move at the same velocity. Understanding this helps in comprehending why perfectly inelastic collisions are crucial in scenarios where bodies do not separate after impact.

Elastic Collision

Elastic collisions stand out due to their conservation of both momentum and kinetic energy. In such a collision, the colliding objects rebound off each other without any loss of energy to sound, heat, or deformation. Hence, the relative speed of separation is equivalent to the relative speed of approach.

The coefficient of restitution \( \epsilon \) for a perfectly elastic collision equals 1. This means that the total kinetic energy and relative velocities remain unchanged before and after the collision. Although perfect elastic collisions are idealized scenarios, they are often used to model real-world phenomena where minimal energy loss occurs, such as in atomic or subatomic particles.

Bounce Height Calculation

Calculating the bounce height after a collision involves understanding the energy conservation principles. When an object, such as a ball, is dropped from a certain height \( h \), it gains kinetic energy just before impact. Upon striking a surface, it rebounds, reaching a new height \( H_1 \).

The relationship between \( H_1 \) and \( h \) can be expressed using the coefficient of restitution \( \epsilon \). The bounce height is given by \( \epsilon = \sqrt{ \frac{H_1}{h} } \). This formula denotes how much of the original height is regained after rebounding. Thus, understanding this helps us compute successive bounce heights in a logical and structured way.

Energy Conservation in Collisions

Energy conservation plays a vital role when analyzing collisions. Not all initial energy remains in the original form post-collision. While energy is neither created nor destroyed, its form can change significantly during collisions.

In perfectly elastic collisions, both kinetic energy and momentum are conserved. For inelastic collisions, however, only momentum remains conserved, as kinetic energy transforms into other forms like heat or sound. This understanding aids in predicting outcomes of collisions and solving related physics problems with greater accuracy.

Physics Problem-Solving

Tackling physics problems involving collisions requires a methodical approach and understanding key concepts, such as momentum, energy conservation, and the coefficient of restitution. To effectively solve these problems, it helps to:

• Clearly define what's conserved and what's not, depending on the type of collision.
• Use equations that relate initial and final velocities and heights to the coefficient of restitution.
• Apply consistent units and accurate calculations, ensuring all terms correlate correctly.

By adhering to these steps, even complex collision problems become manageable. Delving into practice and applying theoretical knowledge is essential for mastering these concepts.