I am deeply concerned about a problem I have in past time even not noticed and slid past it without paying any attention, accepting what books write without the slightest idea that this can be a problem.
According to Newton’s second law if a force $F $ is applied to a body (pay attention! – there is no any option weather it is moving or not) it gets immediately accelerated which means that its speed $v $ changes in the direction of the force immediately after $F$ is applied e.g. after $dt $ – $v\rightarrow v+dv$
That would mean that if $F$ is perpendicular to $v $ the speed $dv$ is also perpendicular to $v$-> which means that $v $ increases in magnitude (not only change of direction) and gets $v+dv $ (or Euclidean geometry is not true in the limit of infinitesimal values which is absurd in the Newtonian space). But this means that work is done by the force and the kinetic energy increases.
If someone insists that nevertheless adding infinitesimal $dv $ doesn’t change the magnitude of $v $ e.g $v+dv=v$ then the same would apply even when $v $ and $dv $ are collinear e.g. adding $dv $ to $v $ doesn’t change the magnitude of $v $ at all.
In textbooks it is stated that $F $ (when perpendicular to $v$) only changes the direction of $v $ which means kinetic energy doesn’t change.
What is wrong in my considerations?