Relativistic momentum via rest mass and velocity
================================================

Momentum (amount of motion) is a vector physical quantity that is a measure of the mechanical
movement of a body. The relativistic momentum also takes into account speed limits equal to the
speed of light.

**Notation:**

#. :math:`c` (:code:`c`) is :attr:`~symplyphysics.quantities.speed_of_light`.

**Notes:**

#. To find rest mass refer to the :ref:`scalar law <Relativistic momentum via rest mass and
   speed>`.

**Links:**

#. `Wikipedia, derivable from here <https://en.wikipedia.org/wiki/Mass_in_special_relativity#Relativistic_mass>`__.

.. py:currentmodule:: symplyphysics.laws.relativistic.vector.relativistic_momentum

.. py:data:: momentum

    Vector of the body's relativistic :attr:`~symplyphysics.symbols.classical_mechanics.momentum`.

Symbol:
    :code:`p`

Latex:
    :math:`{\vec p}`

Dimension:
    :code:`momentum`


.. py:data:: rest_mass

    :attr:`~symplyphysics.symbols.relativistic_mechanics.rest_mass` of the body.

Symbol:
    :code:`m_0`

Latex:
    :math:`m_{0}`

Dimension:
    :code:`mass`


.. py:data:: velocity

    Vector of the body's velocity. See :attr:`~symplyphysics.symbols.classical_mechanics.speed`.

Symbol:
    :code:`v`

Latex:
    :math:`{\vec v}`

Dimension:
    :code:`velocity`


.. py:data:: momentum_law

    :code:`p = m_0 * v * (sqrt(1 - dot(v, v) / c^2))^(-1)`


    Latex:
        .. math::
            {\vec p} = m_{0} {\vec v} \left(\sqrt{1 - \frac{\left( {\vec v}, {\vec v} \right)}{c^{2}}}\right)^{-1}


.. py:data:: velocity_law

    :code:`v = p * c * (sqrt((m_0 * c)^2 + dot(p, p)))^(-1)`


    Latex:
        .. math::
            {\vec v} = {\vec p} c \left(\sqrt{\left(m_{0} c\right)^{2} + \left( {\vec p}, {\vec p} \right)}\right)^{-1}