Momentum

Definition: (Linear) Momentum

The (linear) momentum of a point mass $m$ is the product of its mass times its velocity:

$$m\mathbf{v}$$

The (total) (linear) momentum of a system of point masses $m_1,\dots ,m_n$ is the sum of the momenta of each point mass $m_i$:

$$\sum _{i=1}^n{\mathbf{p}}_i=\sum _{i=1}^n{m}_i{\mathbf{v}}_i$$

The (total) (linear) momentum of a continuous mass distribution $\rho$ with volume $V$ is the integral

$${\int }_V\rho (\mathbf{r})\mathbf{v}(\mathbf{r})\mathrm{dV},$$

where $\mathbf{v}(\mathbf{r})$ is the velocity of the infinitesimally small point mass located at $\mathbf{r}$.

The (total) (linear) momentum of a physical system is the sum of the momenta of its components.

NOTATION

$$\mathbf{p}\mathbf{p}\vec{p}\mathbf{P}\mathbf{P}\vec{P}$$

Definition: Impulse

The impulse of a physical system between two moments $t_1$ and $t_2>t_1$ is the change in the momentum:

$$\mathbf{p}(t_2)-\mathbf{p}(t_1)$$

NOTATION

$$\mathbf{J}\mathbf{J}\vec{J}$$

Theorem: Momentum and Force

The derivative of the momentum of a point mass $m$ with respect to time is the sum ${\sum }_i{}_{\text{on }m}{\mathbf{F}}_i(t^{\ast })$ of all forces currently acting on $m$:

$$\frac{\mathrm{d}\mathbf{P}}{\mathrm{d}t}(t^{\ast })=\sum _i{}_{\text{on }m}{\mathbf{F}}_i(t^{\ast })$$

Note: Non-Inertial Frames

When working in a non-inertial reference frame, the inertial force must be included in ${\sum }_i{}_{\text{on }m}{\mathbf{F}}_i(t)$.

PROOF

TODO

Theorem: Impulse and Force

The impulse of a point mass $m$ between two moments $t_1$ and $t_2>t_1$ is given by the integral of the sum ${\sum }_i{}_{\text{on }m}{\mathbf{F}}_i(t)$ of all forces acting on $m$ from $t_1$ to $t_2$:

$$\mathbf{J}={\int }_{t_1}^{t_2}\sum _i{}_{\text{on }m}{\mathbf{F}}_i(t)\mathrm{dt}$$

Note: Non-Inertial Frames

When working in a non-inertial reference frame, the inertial force must be included in ${\sum }_i{}_{\text{on }m}{\mathbf{F}}_i(t)$.

PROOF

TODO

Theorem: Conservation of Momentum

In an inertial reference frame, the total momentum of an isolated physical system remains constant.

$$\mathbf{P}=\text{const}\mathbf{J}=0$$

PROOF

TODO