Momentum

Momentum ($\vec{p}$) is the quantity of motion carried by an object:

$$\vec{p}=m\vec{v}$$

It is a vector. Its direction is the direction of Velocity, and its magnitude grows with both Mass and speed.

Why momentum is useful

Momentum is conserved in an isolated system: if no external net force acts, total momentum before an interaction equals total momentum after it.

$$\sum {\vec{p}}_{before}=\sum {\vec{p}}_{after}$$

That makes momentum especially useful for collisions and explosions, where forces between objects may be large and messy but internal to the system.

Newton’s second law can also be written as

$${\vec{F}}_{net}=\frac{d\vec{p}}{dt}$$

For constant mass this reduces to ${\vec{F}}_{net}=m\vec{a}$ in Newton’s Laws of Motion.

Common check

Do not write $\vec{p}=m\dot{v}$ for ordinary momentum: $\dot{v}$ is acceleration. Momentum is $m\vec{v}$.

Unit: Newton-Second or $\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-1}$.

Collision habit

Momentum conservation is a system statement. Choose the objects included in the system, then check whether external impulse is negligible in the direction you care about. In two dimensions, conserve momentum component-by-component; the $x$ and $y$ equations are separate scalar equations from one vector law.