Explanation Given: The mass of the block is m . The initial velocity of block is v 1 . The direction of block at initial is θ 1 . The final velocity of block is v 2 . The direction of block at final is θ 2 . Draw the free body diagram of the system as shown in Figure (1). Write the principle of conservation energy. T 1 + V 1 = T 2 + V 2 (I) Here, T 1 , T 2 is the initial and final kinetic energy and V 1 , V 2 is the initial and final potential energy. Write the principle of conservation of linear momentum. ∑ m v 1 = ∑ m v 2 (II) Here, m is the mass of the block, v is the velocity of the block, and the suffixes 1 , 2 indicates the initial and final stages. Write the formula for kinetic energy. T = 1 2 m v 2 (III) Write the formula for potential energy. V = m g h (IV) Here, g is the acceleration due to gravity and h is the elevation of the block from the datum. Conclusion: Refer Figure (1a). At initial: Refer Equation (III). The kinetic energy of the block is T 1 = 1 2 m v 1 2 . When the block is at the elevation of h from the datum. Refer Equation (IV). The potential energy of the block is V 1 = m g h . At final: Refer Equation (III). The kinetic energy of the block is T 2 = 1 2 m v 2 2 . When the block is at the datum. The potential energy of the block is V 2 = 0 . Calculate the final velocity of the block. Substitute 1 2 m v 1 2 for T 1 , m g h for V 1 , 1 2 m v 2 2 for T 2 and 0 for V 2 in Equation (II). 1 2 m v 1 2 + m g h = 1 2 m v 2 2 + 0 v 2 2 = 2 m × ( 1 2 m v 1 2 + m g h ) v 2 2 = v 1 2 + 2 g h v 2 = v 1 2 + 2 g h Thus, the velocity of the block when it reaches the bottom is v 1 2 + 2 g h

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ISBN: 9780137514724

Author: HIBBELER

Publisher: PEARSON

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Chapter 15, Problem 45P

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The speed (v2) and direction (θ2) of the block when it reaches the bottom:

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Chapter 15, Problem 44P

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The speed ( v 2 ) and direction ( θ 2 ) of the block when it reaches the bottom:

Solution Summary: The author explains the principle of conservation of linear momentum and the suffixes 1,2 indicates the initial and final stages.

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Chapter 15 Solutions