# 7: Work and Kinetic Energy

- Page ID
- 4012

The application of Newton’s laws usually requires solving differential equations that relate the forces acting on an object to the accelerations they produce. Often, an analytic solution is intractable or impossible, requiring lengthy numerical solutions or simulations to get approximate results. In such situations, more general relations, like the work-energy theorem (or the conservation of energy), can still provide useful answers to many questions and require a more modest amount of mathematical calculation. In particular, you will see how the work-energy theorem is useful in relating the speeds of a particle, at different points along its trajectory, to the forces acting on it, even when the trajectory is otherwise too complicated to deal with. Thus, some aspects of motion can be addressed with fewer equations and without vector decompositions.

- 7.0: Prelude to Work and Kinetic Energy
- In this chapter, we discuss some basic physical concepts involved in every physical motion in the universe, going beyond the concepts of force and change in motion. These concepts are work, kinetic energy, and power. We explain how these quantities are related to one another, which will lead us to a fundamental relationship called the work-energy theorem. In the next chapter, we generalize this idea to the broader principle of conservation of energy.

- 7.1: Work
- In physics, work represents a type of energy. Work is done when a force acts on something that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. We first define the increment of work dW done by a force acting through an infinitesimal displacement as the dot product of these two vectors. Then, we can add up the contributions for infinitesimal displacements, along a path between two po

- 7.2: Kinetic Energy
- Kinetic energy related to the forces acting on a body and was referred to as “the energy of motion.” The kinetic energy of a particle is one-half the product of the particle’s mass m and the square of its speed v.

- 7.3: Work-Energy Theorem
- Work-Energy Theorem argues the net work done on a particle equals the change in the particle’s kinetic energy. According to this theorem, when an object slows down, its final kinetic energy is less than its initial kinetic energy, the change in its kinetic energy is negative, and so is the net work done on it. If an object speeds up, the net work done on it is positive.

- 7.4: Power
- The concept of work involves force and displacement; the work-energy theorem relates the net work done on a body to the difference in its kinetic energy, calculated between two points on its trajectory. None of these quantities or relations involves time explicitly, yet we know that the time available to accomplish a particular amount of work is frequently just as important to us as the amount itself.

*Thumbnail: One form of energy is mechanical work, the energy required to move an object of mass \(m\) a distance d when opposed by a force \(F\), such as gravity.*

### Contributors

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).