Newtonian mechanics

Momentum

Momentum p=mv is a vector describing translational motion; its change is related to impulse.

Momentum combines mass and velocity and retains the direction of motion. A slow truck may still be harder to stop than a fast bicycle.

Impulse and momentum

Since $\vec p=m\vec v$, reversing velocity reverses momentum. Newton's second law is $\vec F_{net}=d\vec p/dt$; $m\vec a$ assumes constant mass.

Quick check

When force varies with time, one cannot multiply an instantaneous value by the whole duration unless the force is constant. Impulse is the signed area under a force–time graph; in one dimension the average force is $F_{avg}=Delta p/Delta t$. An air bag increases stopping time, so the same momentum change corresponds to a smaller average force on the passenger.

A useful procedure is to resolve momentum into components: $p_x=mv_x$ and $p_y=mv_y$. For perpendicular motions, do not add momentum magnitudes; add each component, then use the Pythagorean theorem. For example, total components of 6 and 8 kg·m/s give a magnitude of 10 kg·m/s. In SI units, $1$ N·s equals $1$ kg·m/s, a useful dimensional check on the impulse theorem.

An object’s momentum may change over a finite interval according to the net impulse accumulated during that interval. In a vector component, a negative sign indicates the opposite direction, not a negative magnitude. State the reference frame and coordinate direction to avoid ambiguity. A final check should confirm both units and the direction of every component.