Explanation Given data: Mass of the ball ( m ) is 57.5 g, which is written as 0.0575 kg. When the ball dropped from a height of 2.5 m onto a concrete surface, it rebounds to a height of 1.4 m. The ball compresses approximately 6 mm during the impact wit surface. Formula used: Write the expression of percentage of kinetic energy reduced as follows. % K lost = K before − K after K before × 100 (1) Here, K before is the kinetic energy of the ball before it rebounding, and K after is the kinetic energy of the ball after it rebounding. Write the expression for kinetic energy. K = 1 2 m v 2 (2) Here, m is the mass of the object, and v is the velocity (speed). Write the expression for gravitational potential energy for the given system as follows. U g = m g y (3) Here, g is the acceleration due to gravity, which is 9.8 m s 2 , and y is the height. Explanation: The speed of the ball when it hits the concrete surface is considered as the speed of the ball before it rebounding for the given problem (which is determined in the problem 81MSPP). The speed of the ball before rebounding is calculated as follows. Calculation of speed of the ball when it hits the surface (speed of the ball before it rebounding for the given problem): Write the expression for conservation of energy for the given conditions as follows. K i + ( U g ) i = K f + ( U g ) f (4) Here, K i is the initial kinetic energy, ( U g ) i is the initial gravitational potential energy, K f is the final kinetic energy, and ( U g ) f is the final gravitational potential energy. Energy is conserved for the given system. The amount of energy (kinetic and potential energy) at the instant of dropping of the ball is equal to the amount of energy during the impact with the surface. Consider the initial velocity of the ball at the time of dropping as v 0 , which is 0 m s since the drops from the rest. Consider the final velocity of the ball when the ball hits the surface as v 1 , which is the speed of the ball before it rebounding and it is to be determined. Consider the initial height at the time of dropping the ball as y 0 , which is 2.5 m above the surface. Consider the final height of the ball when the ball hits the surface as y 1 , which is 0 m since the ball reaches the surface finally. Calculation of initial kinetic energy to find the speed of the ball before it rebounding: Modify the expression in equation (2) to find the initial kinetic energy of the ball as follows. K i = 1 2 m ( v i ) 2 Substitute 0.0575 kg for m , and 0 m s for v i , K i = 1 2 ( 0.0575 kg ) ( 0 m s ) 2 = 1 2 ( 0.0575 kg ) ( 0 m 2 s 2 ) = 0 kg ⋅ m 2 s 2 = 0 J { ∵ 1 kg ⋅ m 2 s 2 = 1 J } Calculation of final kinetic energy to find the speed of the ball before it rebounding: Modify the expression in equation (2) to find the final kinetic energy of the ball as follows. K f = 1 2 m ( v f ) 2 Substitute 0.0575 kg for m , and v 1 for v f , K f = 1 2 ( 0.0575 kg ) ( v 1 ) 2 = 0.02875 ( v 1 ) 2 kg Calculation of initial gravitational potential energy to find the speed of the ball before it rebounding: Modify the expression in equation (3) to find the initial gravitational potential energy of the ball as follows. ( U g ) i = m g y i Substitute 0.0575 kg for m , 9.8 m s 2 for g , and 2.5 m for y i , ( U g ) i = ( 0.0575 kg ) ( 9.8 m s 2 ) ( 2.5 m ) = 1.40875 kg ⋅ m 2 s 2 = 1.40875 J Calculation of final gravitational potential energy to find the speed of the ball before it rebounding: Modify the expression in equation (3) to find the final gravitational potential energy of the ball as follows

College Physics: A Strategic Approach Technology Update, Books a la Carte Plus Mastering Physics with Pearson eText -- Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134201979

Author: Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field

Publisher: PEARSON

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Chapter 10, Problem 84MSPP

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To find: The percentage of decreased kinetic energy.

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Chapter 10, Problem 83MSPP

Chapter 10, Problem 85MSPP

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Chapter 10 Solutions

College Physics: A Strategic Approach Technology Update, Books a la Carte Plus Mastering Physics with Pearson eText -- Access Card Package (3rd Edition)

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The percentage of decreased kinetic energy.

Solution Summary: The author explains the formula used to calculate the percentage of decreased kinetic energy.

To find: The percentage of decreased kinetic energy.

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Chapter 10 Solutions