Maximum height of body thrown at angle to horizon
=================================================

The maximum height of a projectile above its launch position is a function of
the vertical component of the initial velocity, the angle between the initial
velocity and the horizon, and the magnitude of the free fall acceleration.

**Notation:**

#. :math:`g` (:code:`g`) is :attr:`~symplyphysics.quantities.acceleration_due_to_gravity`.

**Conditions:**

#. :math:`\mathbf{v}_0 \cdot \mathbf{g} < 0`, i.e. the vector of initial velocity
   :math:`\mathbf{v}_0` must have a non-zero component that is antiparallel to the
   vector of free fall acceleration :math:`\mathbf{g}`.

#. The height is measured with respect to the horizontal plane where the projectile
   was located at initial time.

**Links:**

#. `Physics LibreTexts. Projectile Motion, Maximum Height (3.3.14) <https://phys.libretexts.org/Bookshelves/University_Physics/Physics_(Boundless)/3%3A_Two-Dimensional_Kinematics/3.3%3A_Projectile_Motion>`__.

.. py:currentmodule:: symplyphysics.laws.gravity.maximum_height_of_a_body_thrown_at_an_angle_to_horizon

.. py:data:: height

    Maximum :attr:`~symplyphysics.symbols.classical_mechanics.height` of the projectile.

Symbol:
    :code:`h`

Latex:
    :math:`h`

Dimension:
    :code:`length`


.. py:data:: initial_speed

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

Symbol:
    :code:`v_0`

Latex:
    :math:`v_{0}`

Dimension:
    :code:`velocity`


.. py:data:: angle

    :attr:`~symplyphysics.symbols.basic.angle` between the initial velocity and the horizon.

Symbol:
    :code:`phi`

Latex:
    :math:`\varphi`

Dimension:
    :code:`angle`


.. py:data:: law

    :code:`h = v_0^2 * sin(phi)^2 / (2 * g)`


    Latex:
        .. math::
            h = \frac{v_{0}^{2} \sin^{2}{\left(\varphi \right)}}{2 g}