5.9 Relativistic energy  (Page 2/16)

When an object is motionless, its speed is $u=0$ and

so that $K_{\text{rel}}=0$ at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical $\frac{1}{2}m{u}^2.$ To show that the expression for $K_{\text{rel}}$ reduces to the classical expression for kinetic energy at low speeds, we use the binomial expansion to obtain an approximation for ${\left(1+\epsilon \right)}^n$ valid for small $\epsilon$ :

by neglecting the very small terms in ${\epsilon }^2$ and higher powers of $\epsilon .$ Choosing $\epsilon =-u^2\text{/}{c}^2$ and $n=-\frac{1}{2}$ leads to the conclusion that γ at nonrelativistic speeds, where $\epsilon =u\text{/}c$ is small, satisfies

A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small speed here, most terms are very small. Thus, the expression derived here for $\gamma$ is not exact, but it is a very accurate approximation. Therefore, at low speed:

Entering this into the expression for relativistic kinetic energy gives

That is, relativistic kinetic energy becomes the same as classical kinetic energy when $u\text{<}\text{<}c.$

It is even more interesting to investigate what happens to kinetic energy when the speed of an object approaches the speed of light. We know that $\gamma$ becomes infinite as u approaches c , so that $K_{\text{rel}}$ also becomes infinite as the velocity approaches the speed of light ( [link] ). The increase in $K_{\text{rel}}$ is far larger than in $K_{\text{class}}$ as v approaches c. An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.

The speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than c always add to less than c . Both the relativistic form for kinetic energy and the ultimate speed limit being c have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach—not reach—the speed of light.

Comparing kinetic energy

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