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Physics · Ch 4 — Work, Energy and Power

Work-Kinetic Energy Theorem

4.2.2

Work-Kinetic Energy Theorem

The work-kinetic energy theorem connects the work done on a body directly to the change produced in its kinetic energy, and can be derived cleanly for the simplest case of a constant force.

Consider a body of mass mm, initially at rest, acted on by a constant force FF that displaces it a distance ss along the direction of the force. The work done is W=FsW = Fs, and by Newton's second law F=maF = ma. Using the (calculus-free) third equation of motion, v2=u2+2asv^2 = u^2 + 2as, we can solve for the acceleration, a=(v2−u2)/(2s)a = (v^2-u^2)/(2s), and substitute into F=maF=ma and then into W=FsW=Fs:

W=Fs=m(v2−u22s)s=12mv2−12mu2W = Fs = m\left(\frac{v^2-u^2}{2s}\right)s = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mu^2

The right-hand side is exactly the change in kinetic energy, ΔKE=KEf−KEi\Delta KE = KE_f - KE_i. So:

W=ΔKE\boxed{W = \Delta KE}

This is the work-kinetic energy theorem: the net work done by the resultant force on a body equals the change in its kinetic energy. Three immediate consequences follow directly:

  1. If the net work done on a body is positive, its kinetic energy increases -- it speeds up. …
Figure 4.7Demonstration of kinetic energy

What this figure shows. A hammer resting motionless on top of a nail exerts no push on the nail and drives it no distance into the wood, because a stationary hammer has no kinetic energy and hence no capacity to do work on the nail. The same hammer, once swung and moving with some speed as it strikes the nail, does drive the nail into the wood, illustrating concretely that only a moving body -- one with kinetic energy -- is able to perform …

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