Work and Kinetic Energy
• When a net force does work on a rigid body, it causes the body's speed to change.

• The work done by the net force is the same as the sum-total of the work done by the action of every force acting the body. If you add up the work done by each of the forces acting on a body you will get the same value as the work done by the net force.

• The work done on a body that caused the body to be set in motion with some speed v can be expressed as function of the body's final speed v and mass m, independent of type of force that acted on the body. We call this function the body's Kinetic Energy.

• The energy a body possesses by virtue of its motion -- its Kinetic Energy -- is not dependent upon how the object reached its state of motion, only upon its current state of motion.

Work Comparison
The work done on three blocks of different mass are compared when the same size force acts over the same distance. Displayed is the transformation of work into kinetic energy of the blocks. Work Comparsion QT Movie

Work-Energy Theorem:

• The energy associated with the work done by the net force does not disappear after the net force is removed (or becomes zero), it is transformed into the Kinetic Energy of the body. We call this the Work-Energy Theorem.

• If the body's speed increases, then the work done on the body is positive and we say its Kinetic Energy has increased. Whereas if the body's speed decreases then it kinetic energy decreases and the change in kinetic energy DKE is negative. In this case the body does positive work on the system slowing it down or alternately the work done on the body is negative.

• If the object is not rigid and any of the forces acting on it deforms the object, then the Work-Energy Theorem will no longer be valid. Some of the energy transferred to the object has gone into deforming the object and is no longer available to increase or decrease the object's Kinetic Energy.

### Derivation of Work-Energy Theorem using a Constant Net Force

Frame of Reference:
Direction of the Net Force.

Next use the Equations for Constant Acceleration that does not involve time:

Calculate the Net Work using the above relationships:

Many problems you encounter related to Work and Energy will have constant forces. While you are trying to learn how to use the concepts of Work and Energy, avoid using Newton's Second Law to solve these problems or else you will have missed the opportunity to learn how to use Work and Energy to solve them. It is reasonable to use the Second Law approach as a way to double-check your Work-Energy solutions.

### Derivation of Work-Energy Theorem using a Variable Net Force

Frame of Reference:
Direction of the Net Force.

To reduce the complexity of the derivation, we will assume that the direction of the Net Force is constant while the work is being done. The Work-Energy Theorem is still valid if the net force changes direction as well as magnitude while the work is being done, provided the body is rigid.

The key step is to convert the calculus definition for acceleration into an expression that is a derivative of x.

Plug this and the Second Law into the definition for Work, and integrate.

Here we have used the relation with x = v to do the integration. Also observe that we have assumed that the body's mass does not change while the force is being applied so that we can remove it from under the integral.