Momentum#

Definition: (Linear) Momentum

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

\[ m\boldsymbol{v} \]

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

\[ \sum_{i=1}^n \boldsymbol{p}_i = \sum_{i=1}^n m_i \boldsymbol{v}_i \]

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

\[ \int_V \rho(\boldsymbol{r}) \boldsymbol{v}(\boldsymbol{r}) \mathop{\mathrm{d}V}, \]

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

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

Notation

\[ \boldsymbol{p} \qquad \mathbf{p} \qquad \vec{p} \qquad \boldsymbol{P} \qquad \mathbf{P} \qquad \vec{P} \]

Definition: Impulse

The impulse of a physical system between two moments \(t_1\) and \(t_2 \gt t_1\) is the change in the momentum:

\[ \boldsymbol{p}(t_2) - \boldsymbol{p}(t_1) \]

Notation

\[ \mathbf{J} \qquad \boldsymbol{J} \qquad \vec{J} \]

Theorem: Momentum and Force

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

\[ \frac{\mathrm{d}\boldsymbol{P}}{\mathrm{d}t}(t^{\ast}) = \sum_i {}_{\text{on }m}\boldsymbol{F}_i(t^{\ast}) \]

Note: Non-Inertial Frames

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

Proof

TODO

Theorem: Impulse and Force

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

\[ \boldsymbol{J} = \int_{t_1}^{t_2} \sum_i {}_{\text{on }m}\boldsymbol{F}_i(t) \mathop{\mathrm{d}t} \]

Note: Non-Inertial Frames

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

Proof

TODO

Theorem: Conservation of Momentum

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

\[ \boldsymbol{P} = \text{const} \qquad \boldsymbol{J} = \boldsymbol{0} \]

Proof

TODO