Velocity relative to reference frame
====================================

For any reference frame, whether it is inertial or not, the motion relative to it can be described using
the position vector relative to that frame's origin.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Velocity#Instantaneous_velocity>`__.

#. `Wikipedia (ru) <https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%BE_%D1%81%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D0%B8_%D1%81%D0%BA%D0%BE%D1%80%D0%BE%D1%81%D1%82%D0%B5%D0%B9#%D0%9E%D0%B1%D1%81%D1%83%D0%B6%D0%B4%D0%B5%D0%BD%D0%B8%D0%B5>`__.

.. py:currentmodule:: symplyphysics.laws.kinematics.vector.velocity_relative_to_reference_frame



.. py:function:: relative_velocity_law(position_, time_)

    Velocity relative to :math:`S`.

    Law:
        :code:`v_rel = Derivative(r(t), t)`

    Latex:
        .. math::
            {\vec v}_\text{rel} = \frac{d \vec r}{d t}

    :param position\_: radius vector, or position vector, of body in :math:`S` as a function of time

        Symbol: :code:`r(t)`

        Latex: :math:`\vec r(t)`

        Dimension: *length*

    :param time\_: time

        Symbol: :code:`t`

        Dimension: *time*

    :return: velocity relative to :math:`S`

        Symbol: :code:`v_rel`

        Latex: :math:`{\vec v}_\text{rel}`

        Dimension: *velocity*


.. py:function:: relative_position_law(initial_position_, velocity_, time_)

    Final position via initial position and velocity as a function of time.

    Law:
        :code:`r = r_0 + Integral(v_rel(t), t)`

    Latex:
        .. math::
            \vec r = {\vec r}_0 + \int {\vec v}_\text{rel}(t) dt

    :param initial\_position\_: position vector in :math:`S` at :math:`t = 0`

        Symbol: :code:`r_0`

        Latex: :math:`{\vec r}_0`

        Dimension: *length*

    :param velocity\_: velocity relative to :math:`S` as a function of time

        Symbol: :code:`v_rel(t)`

        Latex: :math:`{\vec v}_\text{rel}(t)`

        Dimension: *velocity*

    :param time\_: time

        Symbol: :code:`t`

        Dimension: *time*

    :return: position vector in :math:`S` at time :math:`t`

        Symbol: :code:`r`

        Latex: :math:`\vec r`

        Dimension: *length*