Relation for Kinetic energy and the lorentz factor.

[martinhiggs]

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

$$E_{tot} = \gamma m_{0} c^{2}$$

$$E_{tot} = KE + m_{0}c^{2} = \sqrt{p^{2}c^{2} + m_{0}c^{4}}$$

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

The Attempt at a Solution

I thought that the best place to start would be:

$$E_{tot} = KE +m_{0}c^{2}$$

$$KE = E_{tot} - m_{0}c^{2}$$

Also I know that

$$E_{tot} = \gamma m_{0}c^{2}$$

Substituting this in I get:

$$KE = \gamma m_{0}c^{2} - m_{0}c^{2}$$

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!

 

[kreil]

The Lorentz Factor I know is given by,

$$\gamma = \frac{1}{\sqrt{1-\beta^2}}$$ where $\beta = v/c$

At any rate, aren't m0 and c just constants?

 

[diazona]

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 $\gamma$ 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.

 

[martinhiggs]

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!

 

[paranormal]

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!