Points to Ponder

Force does not have to point in the same direction as velocity. It can be along $\vec{v}$, opposite to it, perpendicular to it, or at any other angle. What is always true is that the net force $\vec{F}$ is parallel to the acceleration $\vec{a}$ — that is the real link, not the link to $\vec{v}$.

A body can be momentarily at rest ($v = 0$) and still have a non-zero net force and acceleration. For a ball thrown upward, at the highest point $v = 0$, but the force is still $mg$ downward and the acceleration is $g$ downward. Zero velocity does not imply zero force.

The force on a body at any instant depends only on the current physical situation at that body's location. Force is not a stored quantity that the body carries from its past. When a stone is released from an accelerating train, the moment it leaves the train, the horizontal force from the train vanishes. Only gravity (vertical) remains — there is no "memory" of the train's acceleration.

In Newton's second law, $\vec{F} = m\vec{a}$, the $\vec{F}$ is the net external force from all material agencies acting on the body. The term $m\vec{a}$ is the resulting effect, not a separate force. Never treat $m\vec{a}$ as an extra force alongside $\vec{F}$.

Centripetal force is not a new, independent kind of force. It is simply a label for any real force (tension, gravity, friction, normal reaction, etc.) that points radially inward and causes circular motion. Always identify the actual material agency providing that inward pull.

Static friction is self-adjusting: it can take any value from zero up to its maximum limit $\mu_s N$. The equation $f_s = \mu_s N$ is only valid when the object is just about to slip. Do not use it unless you are certain the maximum static friction is actually acting.

The equation $mg = R$ (weight equals normal reaction) holds only when the body is in vertical equilibrium (no vertical acceleration). In an accelerating lift, $mg$ and $R$ are different. This equality has nothing to do with Newton's third law — it is a consequence of the second law for the specific case of zero acceleration.

In Newton's third law, "action" and "reaction" are simultaneous, mutual forces between two bodies. One does not cause or precede the other. Crucially, they act on different bodies, so they never cancel each other out when considering the net force on a single body.

Terms like friction, normal reaction, tension, air resistance, viscous drag, thrust, buoyancy, weight, and centripetal force are all specific names for forces in different contexts. To avoid confusion, always translate any force into the precise phrase: "force on A by B." …