/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 76 A ball of mass \(m\) is dropped ... [FREE SOLUTION] | 91影视

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Chapter 15: Problem 76

A ball of mass \(m\) is dropped vertically from a height \(h_{0}\) above the ground. If it rebounds to a height of \(h_{1}\), determine the coefficient of restitution between the ball and the ground.

Short Answer

Expert verified
The coefficient of restitution \( e \) is found to be \( e = \sqrt{\frac{h_1}{h_0}} \).

Step by step solution

01

Determine Velocity of the Ball Before Hitting the Ground

Use the principle of conservation of energy. The potential energy of the ball before it falls is transformed into kinetic energy just as it hits the ground. Therefore, \( mgh_0 = \frac{1}{2}mv^2 \). By solving for \( v \), the velocity of the ball just before hitting the ground is \( v = \sqrt{2gh_0} \).
02

Determine the Velocity of the Ball After Rebounding

Apply the conservation of energy principle again as the ball ascends after hitting the ground. The kinetic energy just after hitting is transformed into potential energy at maximum height \( h_1 \). Therefore, \( \frac{1}{2}mu^2 = mgh_1 \). By solving for \( u \), the velocity of the ball right after hitting the ground and moving upwards is \( u = \sqrt{2gh_1} \).
03

Finding the Coefficient of Restitution

The coefficient of restitution \( e \) is defined as the ratio of the speeds after and before an impact, in this case, the rebound of the ball: \( e = \frac{u}{v} \). By substituting the previous found values for \( u \) and \( v \), the coefficient of restitution \( e \) is found to be \( e = \sqrt{\frac{h_1}{h_0}} \).

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