# Kinetic Energy and Work-Energy Theorem

## The work-energy theorem states that the work done by all forces acting on a particle equals the change in the particle's kinetic energy.

#### Key Points

• The work W done by the resultant force on a particle equals the change in the particle's kinetic energy Ek: <equation contenteditable="false">$W=\Delta E_k=\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2$.

• The work-energy theorem can be derived from Newton's second law.

• Work transfers energy from one place to another or one form to another. In more general systems than the particle system mentioned here, work can change the potential energy of a mechanical device, the heat energy in a thermal system, or the electrical energy in an electrical device.

#### Terms

• A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb)

#### Figures

1. ##### Kinetic Energy

A force does work on the block. The kinetic energy of the block increases as a result by the amount of work. This relationship is generalized in the work-energy theorem.

Work in general transfers energy from one place to another or one form to another, and the following is an explanation of the work-energy principle as it applies to particle dynamics. It should be noted that in more general systems, work can change the potential energy of a mechanical device, the heat energy in a thermal system, or the electrical energy in an electrical device.

## The Work-Energy Theorem

The principle of work and kinetic energy (also known as the work-energy theorem) states that the work done by the sum of all forces acting on a particle equals the change in the kinetic energy of the particle (Figure 1). This definition can be extended to rigid bodies by defining the work of the torque and rotational kinetic energy.

The work W done by the resultant (net) force on a particle equals the change in the particle's kinetic energy Ek:

$W=\Delta E_k=\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2$

where vi and vf are the speeds of the particle before and after the change and m is the particle's mass.

## Derivation

For the sake of simplicity, we will consider the case in which the resultant force F is constant in both magnitude and direction and is parallel to the velocity of the particle. The particle is moving with constant acceleration a along a straight line. The relationship between the net force and the acceleration is given by the equation F = ma (Newton's second law), and the particle's displacement d can be expressed by the equation:

$d = \frac{v_2^2 - v_1^2}{2a}$

which follows from the following kinematic equation:

$v_2^2 = v_1^2 + 2ad$

The work of the net force is calculated as the product of its magnitude and the particle's displacement. Substituting the above equations yields:

\begin{align} W &= Fd = mad \\ &= ma \left(\frac{v_2^2 - v_1^2}{2a}\right) = \frac{mv_2^2}{2} - \frac{mv_1^2}{2} \\ & = \Delta {E_k} \end{align}.

#### Key Term Glossary

acceleration
The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).
##### Appears in these related concepts:
Acceleration
the rate at which the velocity of a body changes with time
##### Appears in these related concepts:
displacement
A vector quantity that denotes distance with a directional component.
##### Appears in these related concepts:
Displacement
The length and direction of a straight line between two objects.
##### Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
equation
An assertion that two expressions are equal, expressed by writing the two expressions separated by an equal sign; from which one is to determine a particular quantity.
##### Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
##### Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
##### Appears in these related concepts:
heat
energy transferred from one body to another by thermal interactions
##### Appears in these related concepts:
kinematic
of or relating to motion or kinematics
##### Appears in these related concepts:
kinematics
The branch of mechanics concerned with objects in motion, but not with the forces involved.
##### Appears in these related concepts:
kinetic
Of or relating to motion
##### Appears in these related concepts:
kinetic energy
The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.
##### Appears in these related concepts:
Kinetic Energy
The energy associated with a moving particle or object having a certain mass.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
magnitude
A number assigned to a vector indicating its length.
##### Appears in these related concepts:
mass
The quantity of matter which a body contains, irrespective of its bulk or volume. It is one of four fundamental properties of matter. It is measured in kilograms in the SI system of measurement.
##### Appears in these related concepts:
net force
The combination of all the forces that act on an object.
##### Appears in these related concepts:
parallel
An arrangement of electrical components such that a current flows along two or more paths.
##### Appears in these related concepts:
particle
A very small piece of matter, a fragment; especially, the smallest possible part of something.
##### Appears in these related concepts:
potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
##### Appears in these related concepts:
potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
##### Appears in these related concepts:
resultant
A vector that is the vector sum of multiple vectors
##### Appears in these related concepts:
rigid
Stiff, rather than flexible.
##### Appears in these related concepts:
rigid body
An idealized solid whose size and shape are fixed and remain unaltered when forces are applied; used in Newtonian mechanics to model real objects.
##### Appears in these related concepts:
torque
A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb)
##### Appears in these related concepts:
Torque
Something that produces or tends to produce torsion or rotation; the moment of a force or system of forces tending to cause rotation.
##### Appears in these related concepts:
velocity
A vector quantity that denotes the rate of change of position with respect to time, or a speed with a directional component.
##### Appears in these related concepts:
Velocity
The rate of change of displacement with respect to change in time.
##### Appears in these related concepts:
work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.