<< Chapter < Page | Chapter >> Page > |
Relativistic kinetic energy of any particle of mass m is
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 $\frac{1}{2}\phantom{\rule{0.2em}{0ex}}m{u}^{2}.$ 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.
No object with mass can attain 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.
Notification Switch
Would you like to follow the 'University physics volume 3' conversation and receive update notifications?