Relativistic kinetic energy and force.

[granpa]

classically, an electron accelerating from rest in a uniform electric field will have a kinetic energy proportional to the distance 'd' from its point of origin.

will this continue to hold even when the electron is moving at relativistic velocity?

I understand that the formula for relativistic kinetic energy is

so basically kinetic energy is proportional to gamma - 1

so gamma(d) ≡ d + 1?

 

[granpa]

but according to

http://en.wikipedia.org/wiki/List_of_relativistic_equations

kinetic energy and force in special relativity can not be written exactly like their classical analogues by only replacing the mass with the relativistic mass.

 

[Bill_K]

granpa, There are several ways of writing the force and acceleration, but one thing you can rely on, even in relativity, is energy conservation. The total energy of a charge particle in an E field is E = γmc² + eΦ where Φ is the electrostatic potential, and the value of this will be the same both when the particle is at rest and when it is moving relativistically. (This ignores only the radiation it will emit.)

 

[granpa]

Thank you. Thats was very helpful.

I've also been told that the rate of change of rapidity with respect to proper time will be constant.
I guess that would be the proper acceleration.

gamma(x) = x + 1

a = dg(x)/dx = 1

 

[granpa]

if instead of a uniform electric field we use an inverse square law

F = 1/r^2

KE = integral F dr = 1/r

KE = gamma(r) - 1 = 1/r

gamma(r) = 1 + 1/r