Relation for Kinetic energy and the lorentz factor.

In summary, the conversation discusses finding a relation for kinetic energy as a function of the Lorentz factor, where the kinetic energy can only depend on the Lorentz factor or a constant. The Lorentz factor is defined as 1/sqrt(1-beta^2) and the problem asks to express kinetic energy in terms of gamma and fundamental constants only. However, this is not possible as mass is a necessary variable to distinguish different particles.
  • #1
martinhiggs
24
0

Homework Statement



I have to find a relation for kinetic energy as a function of the lorentz factor, KE(gamma). It can only depend on the lorentz factor or on a constant.

Homework Equations



[tex]E_{tot} = \gamma m_{0} c^{2} [/tex]

[tex]E_{tot} = KE + m_{0}c^{2} = \sqrt{p^{2}c^{2} + m_{0}c^{4}}[/tex]

[tex]\gamma = \sqrt{1 + \frac{v^{2}}{c^2}}[/tex]

The Attempt at a Solution



I thought that the best place to start would be:

[tex]E_{tot} = KE +m_{0}c^{2}[/tex]

[tex]KE = E_{tot} - m_{0}c^{2}[/tex]

Also I know that

[tex]E_{tot} = \gamma m_{0}c^{2}[/tex]

Substituting this in I get:

[tex] KE = \gamma m_{0}c^{2} - m_{0}c^{2} [/tex]

I'm now not sure how to carry on, to get rid of the masses from the equation. Everything I try to do to remove them causes me to have another variable, like energy to then get rid of.

Any suggestions, pointers or help would be greatly appreciated, I've been stuck on this problem for 12 hours now!
 
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  • #2
The Lorentz Factor I know is given by,

[tex] \gamma = \frac{1}{\sqrt{1-\beta^2}}[/tex] where [itex] \beta = v/c[/itex]

At any rate, aren't m0 and c just constants?
 
  • #3
kreil's right. There has to be some property of the particle involved, like mass or energy, because otherwise what would distinguish, say, an electron from a proton? They could both have the same Lorentz factor but their kinetic energies would be vastly different.

If the problem really asks you to find an expression for the kinetic energy in terms of [itex]\gamma[/itex] and fundamental constants only (like the speed of light), then you can go right back to your instructor and tell him/her that it's an impossible problem.
 
  • #4
Sorry, I meant 1/... for the Lorentz factor, typed it out wrong.

Ah, ok, I wasn't thinking of mass as a constant, I see now. Thanks!
 
  • #5


I would approach this problem by first understanding the definitions of kinetic energy and the Lorentz factor.

Kinetic energy (KE) is the energy an object possesses due to its motion. It is given by the equation KE = (1/2)mv^2, where m is the mass of the object and v is its velocity.

The Lorentz factor (γ) is a term used in special relativity to describe the relationship between an object's velocity and its energy. It is given by the equation γ = 1/√(1 - v^2/c^2), where c is the speed of light.

Now, we can see that the Lorentz factor is related to the velocity of an object, while kinetic energy is related to both the mass and velocity of an object. This means that we can express KE in terms of the Lorentz factor by substituting the velocity term in the KE equation with the Lorentz factor equation.

KE = (1/2)m(γc)^2

= (1/2)mγ^2c^2

= (1/2)m(1/(1-v^2/c^2))c^2

= (1/2)m(c^2 - v^2)

= (1/2)m(c^2 - (v/c)^2c^2)

= (1/2)m(c^2 - (γc)^2)

= (1/2)m(c^2 - γ^2c^2)

= (1/2)m(1 - γ^2)c^2

Therefore, the relation for kinetic energy as a function of the Lorentz factor is KE = (1/2)m(1 - γ^2)c^2. This equation shows that kinetic energy is directly proportional to the Lorentz factor, with a factor of (1/2)m(c^2) being the constant term. This means that as the Lorentz factor increases, the kinetic energy of an object also increases, but at a slower rate due to the (1/2)m(c^2) term.

I hope this helps in solving your problem. Remember to always start with a clear understanding of the definitions and equations involved, and then approach the problem step by step. Good luck!
 

Related to Relation for Kinetic energy and the lorentz factor.

1. What is the relation between kinetic energy and the Lorentz factor?

The relation between kinetic energy (KE) and the Lorentz factor (γ) is given by the equation KE = (γ - 1)mc^2, where m is the rest mass of the object and c is the speed of light. This equation shows that as an object's velocity increases, its kinetic energy also increases, and the Lorentz factor takes into account the effects of special relativity.

2. How does the Lorentz factor affect an object's kinetic energy?

The Lorentz factor plays a crucial role in determining an object's kinetic energy, as it takes into account the effects of special relativity. As an object approaches the speed of light, the Lorentz factor increases, causing the object's kinetic energy to increase as well.

3. Can the Lorentz factor be greater than 1?

Yes, the Lorentz factor can be greater than 1. In fact, it approaches infinity as an object reaches the speed of light. This is because the Lorentz factor is defined as γ = 1/√(1 - v^2/c^2), where v is the object's velocity. As v approaches c, the denominator approaches 0, resulting in an infinite Lorentz factor.

4. How is the Lorentz factor used in particle accelerators?

The Lorentz factor is crucial in particle accelerators, as it allows scientists to calculate the energy of accelerated particles. By increasing the velocity of particles close to the speed of light, the Lorentz factor increases, resulting in higher kinetic energies and allowing for the study of high-energy particle interactions.

5. What is the significance of the Lorentz factor in Einstein's theory of relativity?

The Lorentz factor is a fundamental concept in Einstein's theory of special relativity. It is used to describe the effects of time dilation and length contraction at high velocities. The Lorentz factor also plays a role in the famous equation E=mc^2, which relates an object's energy (E) to its mass (m) and the speed of light (c). This concept is essential in our understanding of the universe and has been confirmed through numerous experiments and observations.

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