Kinetic Energy of a Particle. Work-Energy Theorem

Kinetic energy is a scalar measure of the mechanical motion of a particle. Unlike linear momentum, it does not depend on the direction of velocity.

Kinetic energy of a particle

For a particle of mass $m$ moving at speed $v$:

$$T=\frac{mv^2}{2}.$$

Kinetic energy is nonnegative and is measured in joules in SI.

Differential form of the theorem

From $m\vec a=\sum\vec F$ and $d\vec r=\vec vdt$:

$$dT=\sum\vec F\cdot d\vec r=\sum dA.$$

Thus, the elementary change in kinetic energy equals the sum of the elementary works of the forces.

Integral form

Between positions 1 and 2:

$$T_2-T_1=\sum A_{1\to2}.$$

This is the work-energy theorem for a particle.

Advantages of the energy method

The method directly relates speed and position and often avoids solving for time. Reactions of ideal fixed constraints that are perpendicular to displacement do no work and therefore need not appear in the energy equation.

Sign of work and change of speed

Positive net work increases kinetic energy, while negative net work decreases it. However, because kinetic energy depends on $v^2$, the theorem determines speed rather than the direction of the velocity vector.

Example

A 2 kg body starts from rest and the total work of all forces over a displacement is 36 J. Then $mv^2/2=36$, giving $v=6$ m/s.

Common mistakes

• assigning kinetic energy a sign based on velocity direction;
• including the work of only one force instead of the total work;
• confusing the work-energy theorem with conservation of mechanical energy;
• using the energy equation to determine the velocity direction directly.