Ch 09: Work and Kinetic Energy

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 09: Work and Kinetic Energy

Problem 48

Chapter 9, Problem 48

A ball shot straight up with kinetic energy K₀ reaches height h. What height will it reach if the initial kinetic energy is doubled?

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Identify the principle of energy conservation: The total mechanical energy of the ball remains constant, assuming no air resistance. The initial kinetic energy is converted into gravitational potential energy at the maximum height.

Write the expression for the initial kinetic energy and gravitational potential energy. The initial kinetic energy is \( K_0 = \frac{1}{2}mv^2 \), and the gravitational potential energy at height \( h \) is \( U = mgh \), where \( m \) is the mass of the ball, \( g \) is the acceleration due to gravity, and \( h \) is the height.

Relate the initial kinetic energy to the height reached: At the maximum height, all the kinetic energy is converted into potential energy. Thus, \( K_0 = mgh \). Solve for \( h \): \( h = \frac{K_0}{mg} \).

Now consider the case where the initial kinetic energy is doubled: The new kinetic energy is \( 2K_0 \). Using the same energy conservation principle, the new height \( h' \) is given by \( h' = \frac{2K_0}{mg} \).

Compare the new height to the original height: Substitute \( h = \frac{K_0}{mg} \) into the expression for \( h' \). This gives \( h' = 2h \). Therefore, the ball will reach twice the original height if the initial kinetic energy is doubled.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinetic Energy

Kinetic energy (K) is the energy an object possesses due to its motion, calculated using the formula K = 1/2 mv², where m is mass and v is velocity. In the context of the question, the initial kinetic energy K₀ determines how high the ball can ascend before its energy is converted to gravitational potential energy.

Gravitational Potential Energy

Gravitational potential energy (U) is the energy stored in an object as it is lifted against the force of gravity, given by U = mgh, where m is mass, g is the acceleration due to gravity, and h is height. As the ball rises, its kinetic energy is transformed into potential energy until it reaches its maximum height.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the initial kinetic energy of the ball is converted into gravitational potential energy at its peak height, allowing us to relate changes in kinetic energy to changes in height.

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