An electron with initial speed 
is brought to rest by a retarding force of the form
where K is a constant and t is time. The force is applied at t=0.
Calculate the time taken to bring the electron to rest:
(i) Neglecting the effects of radiation reaction.
(ii) Including the ffects of radiation reaction according to the Abraham-Lorentz theory.
Comment on the magnitude of the radiation reaction effect.
SOLUTION:
(i) Neglecting radiation reaction force, the acceleration is:
At t=0 
so 
So the time taken to come to rest neglecting radiation reaction force is:
(ii) To include the effects of radiation reaction we calculate the acceleration using the integral form of the solution to the Lorentz-Abraham equation of motion of the electron.
and 
Perform the integration using:
Integrating to find the velocity as a function of time:
At t=0
So the time T taken to bring the electron to rest (u=0) is the solution to:
Since T must be positive only the positive solution is acceptable.
Assuming the 
term is 
and using the binomial theorem approximation:
Unless K is so large as to invalidate the non-relativistic motion assumption the
term will be 
the 
term and:
So 
is the first order radiation correction for the time to bring the electron
to rest.