7.2 Kinetic energy

It’s plausible to suppose that the greater the velocity of a body, the greater effect it could have on other bodies. This does not depend on the direction of the velocity, only its magnitude. At the end of the seventeenth century, a quantity was introduced into mechanics to explain collisions between two perfectly elastic bodies, in which one body makes a head-on collision with an identical body at rest. The first body stops, and the second body moves off with the initial velocity of the first body. (If you have ever played billiards or croquet, or seen a model of Newton’s Cradle, you have observed this type of collision.) The idea behind this quantity was related to the forces acting on a body and was referred to as “the energy of motion.” Later on, during the eighteenth century, the name kinetic energy    was given to energy of motion.

With this history in mind, we can now state the classical definition of kinetic energy. Note that when we say “classical,” we mean non-relativistic, that is, at speeds much less that the speed of light. At speeds comparable to the speed of light, the special theory of relativity requires a different expression for the kinetic energy of a particle, as discussed in Relativity .

Since objects (or systems) of interest vary in complexity, we first define the kinetic energy of a particle with mass m .

We then extend this definition to any system of particles by adding up the kinetic energies of all the constituent particles:

Note that just as we can express Newton’s second law in terms of either the rate of change of momentum or mass times the rate of change of velocity, so the kinetic energy of a particle can be expressed in terms of its mass and momentum $(\overrightarrow{p}=m\overrightarrow{v}),$ instead of its mass and velocity. Since $v=p\text{/}m$ , we see that

also expresses the kinetic energy of a single particle. Sometimes, this expression is more convenient to use than [link] .

The units of kinetic energy are mass times the square of speed, or $\text{kg}·{\text{m}}^2{\text{/s}}^2$ . But the units of force are mass times acceleration, $\text{kg}·{\text{m/s}}^2$ , so the units of kinetic energy are also the units of force times distance, which are the units of work, or joules. You will see in the next section that work and kinetic energy have the same units, because they are different forms of the same, more general, physical property.