This confusion is eliminated if, at some point, the student is shown how N2
can always be resolved along two mutually perpendicular directions: 1) along
the instantaneous velocity direction, and 2) towards the center of curvature
of the present trajectory. This yields the perfectly general analysis: F_t
= m dv/dt, and F_c = (mv^2)/R, where F_t is the tangential component of the
net force on the particle, F_c is the "centripetal" component of the net
force on the particle, v is the particle's speed, m is its mass, and R is
the current radius of curvature of the particle's trajectory.
Thus N2 is given the simple, perfectly general de-composition:
1) The tangential component of the net force affects only the particle's
speed, not its direction - it speeds up or slows down the particle - it
affects only the magnitude of the velocity vector at any instant.
2) The centripetal component of the net force affects only the particle's
direction, not its speed - it deflects (turns) the particle's direction from
a straight line path at any instant - it changes only the direction of the
particle's path - it affects only the direction of the velocity vector.
This is a powerful and long reaching view. It is useful to do a standard
problem (eg the 2D parabolic trajectory) by applying N2 along these
(changing) directions, instead of along (fixed) horizontal and vertical
directions.
Calculus based classes should be shown that this is a special case of:
Given the vector A and its time derivative dA/dt (another vector):
1) The vector dA/dt can always be resolved into components along, and
perpendicular to, the present direction of the vector A.
2) The component of the vector dA/dt which is perpendicular to the present
direction of the vector A affects only the direction of A, and not its
magnitude.
3) The component of the vector dA/dt which is parallel/anti-parallel to the
present direction of A affects only the direction of the vector A and not
its magnitude, at any instant.
(I think I expressed this better in an earlier, similar discussion, but
here's how it comes out now - in my aging senility.)