Falling body displacement
=========================

Suppose a reference frame :math:`S'` is fixed to a moving object :math:`A` (e.g., Earth) and some
body :math:`B` moving freely (i.e. the sum of external non-gravitational forces acting on it is
zero). In the case of an inertial frame of reference, the displacement of body :math:`B` from the
starting position would follow the usual rule :math:`\vec s = {\vec v}_0 t + \frac{1}{2} {\vec g} t^2`.
But in the case of non-inertial frames, we have to take the Coriolis and the centrifugal force
into account as well, which results into the series shown below.

**Notes:**

#. Note that the series is truncated at the fifth term. More terms can be obtained by plugging the
   result into the equation of motion :math:`\vec a = \vec g + \left[ \vec v, \vec \omega \right]`
   and integrating it over time.

**Conditions:**

#. The sum :math:`\vec F` of all other, non-gravitational forces acting on body :math:`B` is
   :math:`0`.
#. :math:`\vec g` is independent of coordinates.

..
    TODO: add link to source
    TODO: add `O(t^5)` to the law

.. py:currentmodule:: symplyphysics.laws.gravity.vector.falling_body_displacement

.. py:data:: displacement

    Vector of the displacement of body :math:`B` in frame :math:`S'`. See :attr:`~symplyphysics.symbols.classical_mechanics.distance`.

Symbol:
    :code:`s`

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

Dimension:
    :code:`length`


.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time`.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: initial_velocity

    Vector of the initial velocity of body :math:`B`. See :attr:`~symplyphysics.symbols.classical_mechanics.speed`.

Symbol:
    :code:`v_0`

Latex:
    :math:`{\vec v}_{0}`

Dimension:
    :code:`velocity`


.. py:data:: angular_velocity

    Pseudovector of the angular velocity of body :math:`B`. See :attr:`~symplyphysics.symbols.classical_mechanics.angular_speed`.

Symbol:
    :code:`w`

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

Dimension:
    :code:`angle/time`


.. py:data:: acceleration_due_to_gravity

    Vector of the acceleration due to gravity of body :math:`B`.

Symbol:
    :code:`g`

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

Dimension:
    :code:`acceleration`


.. py:data:: law

    :code:`s = v_0 * t + t^2 * (g / 2 + cross(v_0, w)) + t^3 / 3 * (cross(g, w) + 2 * cross(cross(v_0, w), w)) + t^4 / 6 * cross(cross(g, w), w)`


    Latex:
        .. math::
            {\vec s} = {\vec v}_{0} t + t^{2} \left(\frac{{\vec g}}{2} + \left[ {\vec v}_{0}, {\vec \omega} \right]\right) + \frac{t^{3}}{3} \left(\left[ {\vec g}, {\vec \omega} \right] + 2 \left[ \left[ {\vec v}_{0}, {\vec \omega} \right], {\vec \omega} \right]\right) + \frac{t^{4}}{6} \left[ \left[ {\vec g}, {\vec \omega} \right], {\vec \omega} \right]