Explanation Introduction: The impulse-momentum theorem for a body is written as, P i + J e x t e r n a l = P f Here, P i is the initial momentum, P f is the final momentum and J e x t e r n a l is the impulse on the ball due to an external agency. The expression for impulse due to an external agency is written as, J e x t e r n a l = F e x t e r n a l Δ t Here, F e x t e r n a l is the force exerted by the external agency and Δ t is the time interval for the application of force. Sign convention: Consider all quantities measured in the downward direction as positive and quantities measured in the upward direction as negative. Explanation: Understand that the only external interactions for the system (ball) are the reaction force from the ground on the ball and the impulse from the earth. Consider, the happy ball has a mass m and with initial and final velocities as v h i and v h f , respectively. Draw a schematic diagram depicting the motion of happy ball. Write the impulse-momentum equation for the happy ball as the system. P h i + J h , g r o u n d + J h , e a r t h = P h f Here, P h i is the initial momentum of the happy ball and P h f is the final momentum of the happy ball. J h , g r o u n d is the impulse force due to the normal reaction when the happy ball hits the ground and J h , e a r t h is the impulse force due to the force of gravity acting on the body. Since momentum is the product of mass and velocity, substitute m v h i for P h i and m v h f for P h f , m v h i + J h , g r o u n d + J h , e a r t h = m v h f …… (1) The expression for the impulse due to earth on happy ball can be written as, J h , e a r t h = F e a r t h Δ t Substitute mg for F e a r t h (positive sign is means that the weight acts in the downward direction), J h , e a r t h = ( m g ) Δ t Write the kinematic equation of motion for the happy ball when it reaches the ground after being dropped from a height, h . v h i 2 = u h 2 + 2 g h Since, the happy ball is dropped with no initial velocity, substitute 0 m/s for u h , v h i 2 = ( 0 m/s ) + 2 g h v h i = 2 g h Observe that after rebounding to the same height, the magnitude of velocity will remain the same but its direction would be reversed, that is, v h f is − 2 g h . For the happy ball, a new mathematical expression is developed using impulse-momentum equation. Substitute 2 g h for v h i , − 2 g h for v h f , ( m g ) Δ t for J h , e a r t h in equation (1), m ( 2 g h ) + J h , g r o u n d + ( m g ) Δ t = m ( − 2 g h ) J h , g r o u n d = − [ 2 m ( 2 g h ) + ( m g ) Δ t ] The negative sign implies that, the impulse from the ground acts in the upward direction

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The Helmholtz free energy is a thermodynamic potential used to calculate the measure of useful work from a closed system. This system is kept at constant temperature and volume. German physicist Hermann Helmholtz introduced this concept in 1882.

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Chapter 6, Problem 30P

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The impulse-momentum bar chart representation of a sad ball and a happy ball when both the balls of the same mass fall from a height h, the happy ball rebounds to the same height and the sad ball does not bounce at all, considering the ball as the system.

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Chapter 6, Problem 29P

Chapter 6, Problem 31P

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Ch. 6 - 4. A wagon full of medicine balls is rolling along...Ch. 6 - 5. When can you apply the idea that momentum is...Ch. 6 - 6. Choose an example in which the momentum of a...Ch. 6 - Figure Q6.7 shows the velocity-versus-time graphs...Ch. 6 - Which is a safer car bumper in a collision: one...Ch. 6 - 9. Why does an inflated balloon shoot across a...Ch. 6 - In which situation does the momentum of a tennis...Ch. 6 - 11. A toy car with very low friction wheels and...Ch. 6 - You hold a clay ball above a scale and then drop...Ch. 6 - 13. You hold a rubber ball above a scale and then...Ch. 6 - 14 Two battery-powered fan carts are resting on a...Ch. 6 - 15. According to a report on traumatic brain...Ch. 6 - 16. Jim says that momentum is not a conserved...Ch. 6 - Say five important things about momentum (for...Ch. 6 - Three people are observing the same car. One...Ch. 6 - When would a ball hitting a wall have a greater...Ch. 6 - 20. In the previous question, in which case does...Ch. 6 - 21. 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What is...Ch. 6 - * The same 100-g apple is falling from the tree....Ch. 6 - 12. Why does Earth exert the same impulse during...Ch. 6 - 13. * Van hits concrete support In a crash test, a...Ch. 6 - BIO Force exerted by heart on blood About 80 g of...Ch. 6 - 15. The train tracks on which a train travels...Ch. 6 - 16. ** EST Your friend is catching a falling...Ch. 6 - 17 BIO Traumatic brain injury According to a...Ch. 6 - * A 65-kg astronaut pushes against the inside back...Ch. 6 - 19. * You decide to use your garden hose to wash...Ch. 6 - 20. * An egg rolls off a kitchen counter and...Ch. 6 - 21. ** Proportional reasoning During a collision...Ch. 6 - (a) What force is required to stop a 1500-kg car...Ch. 6 - 23. ** EST You drop a 78-g ball vertically onto a...Ch. 6 - * Air bag force on head The graph in Figure P6.24...Ch. 6 - 25. * Equation Jeopardy 1 Invent a problem for...Ch. 6 - * Equation Jeopardy 2 Invent a problem for which...Ch. 6 - * Two carts (100 g and 150 g) on an air track are...Ch. 6 - * A tennis ball of mass m hits a wall at speed v...Ch. 6 - 29. * A tennis ball traveling at a speed of v...Ch. 6 - Prob. 30P Ch. 6 - Prob. 31P Ch. 6 - * You hold a beach ball above your head and then...Ch. 6 - 33. * A basketball player drops a 0.60-kg...Ch. 6 - * Bar chart Jeopardy Invent a problem for each of...Ch. 6 - * A baseball bat contacts a 0.145-kg baseball for...Ch. 6 - A cannon mounted on the back of a ship fires a...Ch. 6 - A 10-kg sled carrying a 30-kg child glides on a...Ch. 6 - 38. 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Drifting space mechanic An astronaut with a...Ch. 6 - Prob. 55P Ch. 6 - 56. Rocket stages A 5000-kg rocket ejects a...Ch. 6 - 57. * A rocket has just ejected fuel. With the...Ch. 6 - 58. * Car collision A 1180-kg car traveling south...Ch. 6 - 59. * Ice skaters collide While ice skating, you...Ch. 6 - 1015-kg meteorite traveling at about 10 km/s...Ch. 6 - 61. * Three friends play beach volleyball. 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The impulse-momentum bar chart representation of a sad ball and a happy ball when both the balls of the same mass fall from a height h , the happy ball rebounds to the same height and the sad ball does not bounce at all, considering the ball as the system.

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The impulse-momentum bar chart representation of a sad ball and a happy ball when both the balls of the same mass fall from a height h, the happy ball rebounds to the same height and the sad ball does not bounce at all, considering the ball as the system.

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