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The Work-Energy Theorem

Now that we have a definition of work, we can apply the concept to kinematics. Just as force was related to acceleration through F = ma , so is work related to velocity through the Work-Energy Theorem.

Derivation of the Work-Energy Theorem

It would be easy to simply state the theorem mathematically. However, an examination of how the theorem was generated gives us a greater understanding of the concepts underlying the equation. Because a complete derivation requires calculus, we shall derive the theorem in the one-dimensional case with a constant force.

Consider a particle acted upon by a force as it moves from x o to x f . Its velocity also increases from v o to v f . The net work on the particle is given by:

W net = F net(x f - x o)

Using Newton's Second Law we can substitute for F:

W net = ma(x f - x o)

Given uniform acceleration, v f 2 - v I 2 = 2a(x f - x o) . Substituting for a(x f - x o) into our work equation, we find that:

W net = mv f 2 - mv o 2    

This equation is one form of the work-energy equation, and gives us a direct relation between the net work done on a particle and that particle's velocity. Given an initial velocity and the amount of work done on a particle, we can calculate the final velocity. This is important for calculations within kinematics, but is even more important for the study of energy, which we shall see below.

Kinetic Energy and the Work-Energy Theorem

As is evident by the title of the theorem we are deriving, our ultimate goal is to relate work and energy. This makes sense as both have the same units, and the application of a force over a distance can be seen as the use of energy to produce work. To complete the theorem we define kinetic energy as the energy of motion of a particle. Taking into consideration the equation derived just previously, we define the kinetic energy numerically as:

K = mv 2    

Thus we can substitute K in our work energy theorem:

W net = mv f 2 - mv I 2 = K f - K o

Implying that

W net = ΔK    

This is our complete Work-Energy theorem. It is powerfully simple, and gives us a direct relation between net work and kinetic energy. Stated verbally, the equations says that net work done by forces on a particle causes a change in the kinetic energy of the particle.

Though the full applicability of the Work-Energy theorem cannot be seen until we study the conservation of energy, we can use the theorem now to calculate the velocity of a particle given a known force at any position. This capability is useful, since it relates our derived concept of work back to simple kinematics. A further study of the concept of energy, however, will yield far greater uses for this important equation.

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