Excuse me ... .

In sight, it is impossible that a classical particle can be accelerated without a force acting on it ... . In other word ... , the velocity of the particle cannot change if the force acting on it is zero, sightly ... .

But ... , in fact, this opinion is not always true generally ... .

Really, the Newton’s second law of motion is a differential equation

where the

and the

, respectively, are the force and the momentum of the particle as function of the time

... .

If the velocity and mass of the particle, respectively, are

and

as function of time

, then

... . Thus, in general,

where

is rate of the mass, and

is its acceleration ... . The last expression doesn’t only obey in non-relativistic cases, but also in relativistic cases ... .

If we set

, then

... . In this case,

alias

alias

alias

... . Thus (by subtituting or by differentiating), this yields

... . [Here, the

and the

denote the initial mass and the initial velocity, respectively ... .]

In another case, if

, then

... .

Therefore, really, the force merely changes the momentum rather than the velocity ... .