Displacement in critical damping
================================

In the presence of a damping force in the oscillating system, the system's behavior
depends on the value of the damping ratio. When it is equal to :math:`1`, the system returns
to equilibrium as quickly as possible without oscillating, but overshoot can occur if
initial velocity is nonzero. This behavior is also called *critical damping*.

**Conditions:**

#. The system is critically damped, i.e. its damping ratio :math:`\zeta = 1`.

.. py:currentmodule:: symplyphysics.laws.kinematics.damped_oscillations.displacement_in_critical_damping

.. py:data:: displacement

    Displacement from rest, usually a function of time. See :attr:`~symplyphysics.symbols.classical_mechanics.euclidean_distance`.

Symbol:
    :code:`d`

Latex:
    :math:`d`

Dimension:
    :code:`length`


.. py:data:: time

    :attr:`~symplyphysics.symbols.basic.time` at which :attr:`~displacement` is measured.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: undamped_angular_frequency

    :attr:`~symplyphysics.symbols.classical_mechanics.angular_frequency` of the undamped oscillator.

Symbol:
    :code:`w`

Latex:
    :math:`\omega`

Dimension:
    :code:`angle/time`


.. py:data:: initial_position

    Initial :attr:`~symplyphysics.symbols.classical_mechanics.position` of the oscillator.

Symbol:
    :code:`x_0`

Latex:
    :math:`x_{0}`

Dimension:
    :code:`length`


.. py:data:: initial_speed

    Initial :attr:`~symplyphysics.symbols.classical_mechanics.speed` of the oscillator.

Symbol:
    :code:`v_0`

Latex:
    :math:`v_{0}`

Dimension:
    :code:`velocity`


.. py:data:: law

    :code:`d = exp(-w * t) * (x_0 + (v_0 + x_0 * w) * t)`


    Latex:
        .. math::
            d = \exp{\left(- \omega t \right)} \left(x_{0} + \left(v_{0} + x_{0} \omega\right) t\right)