Ch.1 - Introduction: Matter, Energy, and Measurement

Problem 29

Chapter 1, Problem 29

What is the kinetic energy and velocity of the aluminum sphere in Problem 1.4 at the moment it hits the ground? (Assume that energy is conserved during the fall and that 100% of the sphere's initial potential energy is converted to kinetic energy by the time impact occurs.)

Verified step by step guidance

Identify the initial potential energy of the aluminum sphere using the formula: $PE = mgh$, where $m$ is the mass, $g$ is the acceleration due to gravity, and $h$ is the height from which the sphere falls.

Assume that all of the initial potential energy is converted into kinetic energy at the moment of impact. Therefore, set the potential energy equal to the kinetic energy: $KE = \frac{1}{2}mv^2 = mgh$.

Solve for the velocity $v$ by rearranging the kinetic energy equation: $v = \sqrt{2gh}$.

Substitute the known values for $g$ and $h$ into the equation to find the velocity $v$ of the sphere at the moment of impact.

Use the calculated velocity to determine the kinetic energy at impact using the formula: $KE = \frac{1}{2}mv^2$.

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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 is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is the mass and v is the velocity of the object. In the context of the falling aluminum sphere, its kinetic energy at the moment of impact is equal to the potential energy it had at its initial height, assuming no energy losses.

Potential Energy

Potential energy is the stored energy of an object due to its position or configuration, commonly expressed as PE = mgh, where m is mass, g is the acceleration due to gravity, and h is the height above the ground. For the aluminum sphere, its initial potential energy is converted entirely into kinetic energy as it falls, illustrating the principle of conservation of energy.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the potential energy of the aluminum sphere at its height is transformed into kinetic energy as it falls, allowing us to equate the two forms of energy to find the sphere's velocity and kinetic energy at impact.

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Related Practice

Textbook Question

Two positively charged particles are first brought close together and then released. Once released, the repulsion between particles causes them to move away from each other. (a) This is an example of potential energy being converted into what form of energy?

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Textbook Question

Two positively charged particles are first brought close together and then released. Once released, the repulsion between particles causes them to move away from each other. (b) Does the potential energy of the two particles prior to release increase or decrease as the distance between them is increased.

Textbook Question

For each of the following processes, does the potential energy of the object(s) increase or decrease? (b) Water is pumped from ground level to the reservoir of a water tower 30 m above the ground.

Textbook Question

Convert the following expressions into exponential notation: (a) 3 terameters (tm)

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Textbook Question

Convert the following expressions into exponential notation: (b) 2.5 femtoseconds (fs) (c) 57 micrometers (mm)

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Textbook Question

Convert the following expressions into exponential notation: (d) 8.3 megagrams (mg).

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