# 28.6 Relativistic energy  (Page 4/12)

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${W}_{\text{net}}=E-{E}_{0}=\gamma {\mathrm{mc}}^{2}-{\mathrm{mc}}^{2}=\left(\gamma -1\right){\mathrm{mc}}^{2}.$

Relativistically, we have ${W}_{\text{net}}={\text{KE}}_{\text{rel}}$ .

## Relativistic kinetic energy

Relativistic kinetic energy is

${\text{KE}}_{\text{rel}}=\left(\gamma -1\right){\mathrm{mc}}^{2}.$

When motionless, we have $v=0$ and

$\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}=1,$

so that ${\text{KE}}_{\text{rel}}=0$ 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 $\frac{1}{2}{\mathrm{mv}}^{2}$ . To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for $\gamma$ at low velocities gives

$\gamma =1+\frac{1}{2}\frac{{v}^{2}}{{c}^{2}}.$

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 velocity here, most terms are very small. Thus the expression derived for $\gamma$ here is not exact, but it is a very accurate approximation. Thus, at low velocities,

$\gamma -1=\frac{1}{2}\frac{{v}^{2}}{{c}^{2}}.$

Entering this into the expression for relativistic kinetic energy gives

${\text{KE}}_{\text{rel}}=\left[\frac{1}{2}\frac{{v}^{2}}{{c}^{2}}\right]{\mathrm{mc}}^{2}=\frac{1}{2}{\mathrm{mv}}^{2}={\text{KE}}_{\text{class}}.$

So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when $v\text{<<}c$ .

It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that $\gamma$ becomes infinite as $v$ approaches $c$ , so that KE rel also becomes infinite as the velocity approaches the speed of light. (See [link] .) 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

No object with mass can attain the speed of light.

So 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: relativistic energy versus classical kinetic energy

An electron has a velocity $v=0\text{.}\text{990}c$ . (a) Calculate the kinetic energy in MeV of the electron. (b) Compare this with the classical value for kinetic energy at this velocity. (The mass of an electron is $9\text{.}\text{11}×{\text{10}}^{-\text{31}}\phantom{\rule{0.25em}{0ex}}\text{kg}$ .)

Strategy

The expression for relativistic kinetic energy is always correct, but for (a) it must be used since the velocity is highly relativistic (close to $c$ ). First, we will calculate the relativistic factor $\gamma$ , and then use it to determine the relativistic kinetic energy. For (b), we will calculate the classical kinetic energy (which would be close to the relativistic value if $v$ were less than a few percent of $c$ ) and see that it is not the same.

Calculate the Newton's the weight of a 2.5 Kilogram of melon. What is its weight in pound?
calculate the tension of the cable when a buoy with 0.5m and mass of 20kg
what is displacement
it's the time rate of change of distance
Mollamin
distance in a given direction is diplacement
Musa
Distance in a spacified direction
you shouldn't say distance,displacement and distance are two different things .distance can be lopped curved but displacement is always in a straight line so you can't use distance to define it. displacement is the change of position in a specified direction.
Joshua
Well stayed josh👍
Joshua
well explained
Mary
what is the meaning of physics
to study objects in motion and how they interact or take part in the natural phenomenon of the universe.
Phill
an object that has a small mass and an object has a large mase have the same momentum which has high kinetic energy
The with smaller mass
how
Faith
Since you said they have the same momentum.. So meaning that there is more like an inverse proportionality in the quantities used to find the momentum. We are told that the the is a larger mass and a smaller mass., so we can conclude that the smaller mass had higher velocity as compared to other one
Mathamaticaly correct
Mathmaticaly correct :)
I have proven it by using my own values
Larger mass=4g Smaller mass=2g Momentum of both=8 Meaning V for L =2 and V for S=4 Now find there kinetic energies using the data presented
grateful soul...thanks alot
Faith
Welcome
2 stones are thrown vertically upward from the ground, one with 3 times the initial speed of the other. If the faster stone takes 10 s to return to the ground, how long will it take the slower stone to return? If the slower stone reaches a maximum height of H, how high will the faster stone go
30s
how can i calculate it's height
Julliene
is speed the same as velocity
no
Nebil
in a question i ought to find the momentum but was given just mass and speed
Faith
just multiply mass and speed then you have the magnitude of momentem
Nebil
Yes
Consider speed to be velocity
it worked our . . thanks
Faith
Distinguish between semi conductor and extrinsic conductors
Suppose that a grandfather clock is running slowly; that is, the time it takes to complete each cycle is longer than it should be. Should you (@) shorten or (b) lengthen the pendulam to make the clock keep attain the preferred time?
I think you shorten am not sure
Uche
shorten it, since that is practice able using the simple pendulum as experiment
Silvia
it'll always give the results needed no need to adjust the length, it is always measured by the starting time and ending time by the clock
Paul
it's not in relation to other clocks
Paul
wat is d formular for newton's third principle
Silvia
okay
Silvia
shorten the pendulum string because the difference in length affects the time of oscillation.if short , the time taken will be adjusted.but if long ,the time taken will be twice the previous cycle.
discuss under damped
resistance of thermometer in relation to temperature
how
Bernard
that resistance is not measured yet, it may be probably in the next generation of scientists
Paul
Is fundamental quantities under physical quantities?
please I didn't not understand the concept of the physical therapy
physiotherapy - it's a practice of exercising for healthy living.
Paul
what chapter is this?
Anderson
this is not in this book, it's from other experiences.
Paul
am new in the group
Daniel
Sure
What is Boyce law
Boyles law states that the volume of a fixed amount of gas is inversely proportional to pressure acting on that given gas if the temperature remains constant which is: V<k/p or V=k(1/p)