# Kinetic Energy Theorem

Particle of mass *m* moving from point A to point B

We say that a body has kinetic energy (Ec) when it has mass and speed. More specifically, we say that this kinetic energy corresponds to the *work done on the body* . But how could we represent this kinetic energy in terms of * m* and

*?*

**v**Let us consider a particle whose mass is * m* , which passes through a point

**A**with speed ; and by a point

**B**with velocity ; along any trajectory, under the action of any number of forces. We can determine the total work done by these forces between points

**A**and

**B**through the following equation:

Based on this result, it is reasonable to assume that the kinetic energy of a body of mass **m** and speed v is given by:

Thus, we can say that in the first equation the factor *m . v ^{2 }_{B} /2* is the kinetic energy at point B (E

_{cB}); and

*m. v*is the kinetic energy at point A (E

^{2 }_{A}/2_{cA}). In this way, we can write the first equation in another way:

*τ _{AB} = E _{cB} – E _{cA} = ∆E _{c} (change in kinetic energy)*

Therefore, we can say that these equations are mathematical ways of stating the Kinetic Energy Theorem, which says:

*The total work of the forces acting on a particle is equal to the change in the kinetic energy of that particle.*

Note: if this total work is positive, there will be an increase in kinetic energy; if it is negative, there will be a decrease in kinetic energy.