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W net = E E 0 = γ mc 2 mc 2 = γ 1 mc 2 .

Relativistically, we have W net = KE rel size 12{W="KE" rSub { size 8{"rel"} } } {} .

Relativistic kinetic energy

Relativistic kinetic energy is

KE rel = γ 1 mc 2 . size 12{"KE" rSub { size 8{"rel"} } = left (γ - 1 right ) ital "mc" rSup { size 8{2} } } {}

When motionless, we have v = 0 size 12{v=0} {} and

γ = 1 1 v 2 c 2 = 1 , size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } =1} {}

so that KE rel = 0 size 12{"KE" rSub { size 8{"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 1 2 mv 2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } } {} . To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for γ size 12{γ} {} at low velocities gives

γ = 1 + 1 2 v 2 c 2 . size 12{γ=1+ { {1} over {2} } { {v rSup { size 8{2} } } over {c rSup { size 8{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 γ size 12{γ} {} here is not exact, but it is a very accurate approximation. Thus, at low velocities,

γ 1 = 1 2 v 2 c 2 . size 12{γ - 1= { {1} over {2} } { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } {}

Entering this into the expression for relativistic kinetic energy gives

KE rel = 1 2 v 2 c 2 mc 2 = 1 2 mv 2 = KE class .

So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when v << c size 12{v"<<"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 γ size 12{γ} {} becomes infinite as v size 12{v} {} approaches c size 12{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 size 12{c} {} always add to less than c size 12{c} {} . Both the relativistic form for kinetic energy and the ultimate speed limit being c size 12{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.

In this figure a graph is shown on a coordinate system of axes. The x-axis is labeled as speed v (m/s). On the x-axis, velocity of the object is shown in terms of the speed of light starting from zero at origin to c, where c is the speed of light. The y-axis is labeled as Kinetic Energy K E (J). On the y-axis, relativistic kinetic energy is shown starting from 0 at origin to 1.0. The graph K sub r e l of relativistic kinetic energy is concave up and moving upward along the vertical line at x equals c. This graph shows that relativistic kinetic energy approaches infinity as the velocity of an object approaches the speed of light. Also shown is that when the speed of the object is equal to the speed of light c the kinetic energy is known as classical kinetic energy, which is denoted as K E sub class.
This graph of KE rel size 12{"KE" rSub { size 8{"rel"} } } {} versus velocity shows how kinetic energy approaches infinity as velocity approaches the speed of light. It is thus not possible for an object having mass to reach the speed of light. Also shown is KE class size 12{"KE" rSub { size 8{"class"} } } {} , the classical kinetic energy, which is similar to relativistic kinetic energy at low velocities. Note that much more energy is required to reach high velocities than predicted classically.

Comparing kinetic energy: relativistic energy versus classical kinetic energy

An electron has a velocity v = 0 . 990 c size 12{v=0 "." "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 . 11 × 10 31 kg size 12{9 "." "11" times "10" rSup { size 8{ - "31"} } " 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 size 12{c} {} ). First, we will calculate the relativistic factor γ size 12{γ} {} , 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 size 12{v} {} were less than a few percent of c size 12{c} {} ) and see that it is not the same.

Questions & Answers

Calculate the Newton's the weight of a 2.5 Kilogram of melon. What is its weight in pound?
Rialyn Reply
calculate the tension of the cable when a buoy with 0.5m and mass of 20kg
Iga Reply
what is displacement
Nyamza Reply
it's the time rate of change of distance
Mollamin
distance in a given direction is diplacement
Musa
Distance in a spacified direction
Gift
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👍
Gift
thank you gift.
Joshua
well explained
Mary
what is the meaning of physics
Alausa Reply
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
Faith Reply
The with smaller mass
Gift
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
Gift
Mathamaticaly correct
megavado
Mathmaticaly correct :)
megavado
I have proven it by using my own values
Gift
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
Gift
grateful soul...thanks alot
Faith
Welcome
Gift
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
Julliene Reply
30s
Gift
how can i calculate it's height
Julliene
is speed the same as velocity
Faith Reply
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
Gift
Consider speed to be velocity
Gift
it worked our . . thanks
Faith
Distinguish between semi conductor and extrinsic conductors
Okame Reply
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?
Aj Reply
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.
FADILAT
discuss under damped
Prince Reply
resistance of thermometer in relation to temperature
Ifeanyi Reply
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?
Igwe Reply
please I didn't not understand the concept of the physical therapy
John Reply
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
please I have probably with calculate please can you please and help me out
John Reply
Sure
Gift
What is Boyce law
Sly Reply
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)
FADILAT
Practice Key Terms 3

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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