# Kinetic Energy and Momentum of a Relativistic Particle

• Mister T
In summary, Riggs derives two expressions for the kinetic energy of a massive particle, which are useful for teaching relativity in an introductory course.
Mister T
Gold Member
College-level introductory physics textbooks usually devote a chapter to special relativity. Peter J. Riggs in his article appearing in the February 2016 issue of The Physics Teacher (pp 80-82) derives a couple of expressions for the kinetic energy of a massive (as opposed to massless) particle that I find very useful. I don't recall having seen them in those textbooks, and Riggs claims that they rarely do. They are, I think, time-savers for those of us trying to teach relativity in this course.

The first is ##T=\frac{p^2}{(\gamma+1)m}## and the second is ##T=(\frac{\gamma^2}{\gamma+1})mv^2##.

In these expressions ##T## is the kinetic energy, ##p## is the magnitude of the 3-momentum, ##m## is the mass, ##v## is the speed, and ##\gamma## is the relativistic factor defined by $$\gamma=\frac{1}{\sqrt{1-(\frac{v}{c})^2}}.$$
It's easy for students to see that in the low speed limit ##\gamma \approx 1## and that these expression then reduce to the familiar Newtonian versions: ##T=\frac{p^2}{2m}## and ##T=\frac{1}{2}mv^2## respectively. There is therefore no need to carry out the series expansion that is usually done to demonstrate the latter.

Therefore ##p^2c^2=E^2-m^2c^4=(E-mc^2)(E+mc^2)=T(E+mc^2)=T(\gamma mc^2+mc^2)=Tmc^2(\gamma+1)##.

Dividing both sides by ##c^2## gives ##p^2=Tm(\gamma+1)##. Solve for ##T## and you have the first expression. Replace ##p## with ##\gamma mv## and you get the second expression.

Edit: Fixed the missing square root in the expression for ##\gamma##. Thanks!

Last edited:
robphy and PAllen
Riggs' article [thanks @Mister T for making this known]
is available at
A Comparison of Kinetic Energy and Momentum in Special Relativity and Classical Mechanics
Peter J. Riggs
Phys. Teach. 54, 80 (2016)
http://scitation.aip.org/content/aapt/journal/tpt/54/2/10.1119/1.4940169 [appears to be open access, for now]

Along these lines, there is an older article [not referenced by the above]:
Parallels between relativistic and classical dynamics for introductory courses
Donald E. Fahnline
Am. J. Phys. 43, 492 (1975)
http://scitation.aip.org/content/aapt/journal/ajp/43/6/10.1119/1.9775

I think these are useful, as you say, to more easily show the classical limit of the relativistic expressions.. without calculus or a series expansion.
Unfortunately, these relativistic expressions, as derived in these works, don't "fall out" naturally from first principles... but are arranged to resemble the classical limit. So, the physical intuition is limited to being an algebraic expression relating the two expressions.
To be clear... it's very useful and should be better known, but it is limited.

(By the way, you forgot the square-root in the expression for gamma.)

I completely agree with you, these expressions are definitely time-savers when teaching special relativity. It's always great to have alternative ways of presenting important concepts, and I think these derivations are a great addition to the traditional textbook approach.

I also appreciate the reminder of the low speed limit and how these expressions reduce to the familiar Newtonian versions. This will definitely help students make connections between the two theories and understand the significance of special relativity.

Thank you for sharing this information and providing such a clear derivation. I will definitely be incorporating these expressions into my teaching of special relativity.

## 1. What is the difference between kinetic energy and momentum of a relativistic particle?

Kinetic energy is the energy an object possesses due to its motion, while momentum is a measure of an object's inertia and how difficult it is to stop its motion. In the context of relativistic particles, the main difference is that kinetic energy takes into account the particle's mass and its speed, while momentum only considers its speed.

## 2. How is the kinetic energy of a relativistic particle calculated?

The kinetic energy of a relativistic particle is calculated using the formula KE = (γ - 1)mc², where γ is the Lorentz factor and m is the mass of the particle. This formula takes into account the increase in kinetic energy as the particle approaches the speed of light.

## 3. What is the relationship between kinetic energy and momentum in special relativity?

In special relativity, the momentum of a particle is given by p = γmv, where v is the velocity of the particle. This means that as the particle's speed approaches the speed of light, its momentum increases significantly while its kinetic energy approaches infinity.

## 4. How does the concept of relativistic mass affect the calculation of kinetic energy and momentum?

In special relativity, the mass of a particle is not constant and can change with its velocity. This is known as relativistic mass and it affects the calculation of kinetic energy and momentum. As the particle's speed increases, its relativistic mass also increases, resulting in a higher kinetic energy and momentum.

## 5. Can the kinetic energy of a relativistic particle ever exceed its rest energy?

Yes, the kinetic energy of a relativistic particle can exceed its rest energy. This is known as the mass-energy equivalence principle, which states that mass and energy are interchangeable. In special relativity, the kinetic energy of a particle can approach infinity as its speed approaches the speed of light, while its rest energy remains constant.

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