# Module 1 -- Work and Kinetic Energy

#### Learning objectives

After completing this module you should be able to

1. Define the kinetic energy of an object
2. Define the work done by a force
3. Explain why the work done by a force on an object is the responsible of the change of the kinetic energy of the object.

In this module, we explore the consequences of applying a force to an object over a specified distance. Consider a purely one-dimensional situation in which a constant net force acting in the x-direction Fnet,x is applied to a block of mass m with an initial x-velocity vi over a displacement Δx.

It is possible to show (see the derivation below) that the result of this action is:

$\frac{1}{2} mv_{f}^{2} = \frac{1}{2} m v_{i}^{2} + F_{net,x}\; \Delta x$

This equation has the form of a law of change, with 1/2mv2 playing the role of the motion descriptor and Fnet,xΔx accounting for the interactions.

## Kinetic Energy

The new description of motion we have developed here is called kinetic energy,and given the symbol K.

$K = \frac{1}{2} m v^{2}\;$

## Work

The new representation of interactions that we have developed is called work, and given the symbol W. The speed or kinetic energy of the object changes because there is a component of the force along the direction of motion acting over a distance. The work quantifies the change of the kinetic energy produced by a force being applied along a given path between an initial and a final position. In summary,

1. The Work is the cause of the change in the kinetic energy of the object
2. The work depends on the component of the force along the displacement vector
3. The work depends on the initial and final position.

In the example above the force is constant and the path followed by the block is a straight line. Under these conditions the work is the product of the component of the force along the displacement and the magnitude of the displacement vector:

Work done by a Constant Force along a straight line
$W_{f,i} = F_{x}\;\Delta x$

 Note: the subscripts i and f are needed to indicate that the force is being applied between the corresponding initial and final positions.

If the force is not constant or/and the displacement vector is not constant then the work is obtained by an integral:

Work done by a Non - Constant Force or/and a Non - Constant displacement
$W_{f,i} = \int_{\vec{r}_{i}}^{\vec{r}_{f}} F_{r}\;dr$

where Fr is the component of the force along the displacement vector of magnitude dr.

In the figure below a particle of mass m is moving along an arbitrary path from an initial to a final position. A given force of magnitude F is applied on the particle. Between two consecutive instant of time, t and t+dt, the applied force and the displacement vector form an angle θ. The component of the force along the displacement vector is Fr .

The work done by the force along the small distance dr is dW=Frdr = F cosθ dr. Note that dW changes along the path followed by the particle if the magnitude or direction of the force changes with position, or the displacement vector changes with the position.

To obtain the work between the initial and final position, Wi,f, we must integrate dW along the path followed by the particle.

## Work-Kinetic Energy Theorem

The law of change we developed above is sometimes called the work-kinetic energy theorem, and can be written:

$K_{f} = K_{i} + W_{f,i}\;$

## The Units of Work and Energy

The SI unit of energy is the Joule (J). As can be inferred from the equation above, the Joule is equivalent to:

$\mathcal{}1\mbox{ Joule} = 1 \mbox{kg m}^{2}\mbox{/s}^{2} = 1 \mbox{N m}$

## Kinetic Energy is Additive in a Multi-Body System

The total kinetic energy of a system composed of several objects is the sum of the individual kinetic energies of the objects.