Vertical launch is the rectilinear and vertical motion that an object describes when it is only under the gravitational force action, the air resistance is neglected.

Vertical launch. For clarity of the figure, a small horizontal offset that should be considered negligible was added.

A particular case of vertical launch is the free fall, which is the movement of an abandoned object from a certain height.

Free fall

It is the straight and vertical motion of an object when it is only under the action of the gravitational force, without taking into account air resistance. The Earth's gravitational pull attracts objects and has the following characteristics:

Direction
Vertical, passing through the Earth's center of gravity and pointing downwards to the Earth's center.
Module
For motions close to the Earth's surface, the gravity acceleration modulus changes slightly from place to place, but, on average, it is approximately \(9.81 m/s^2 \). Sometimes, is convenient to round to \(10 m/s^2 \) to simplify the calculations.
When the object is dropped, and there is a free fall from a small height \(h << R\) relative to the Earth's radius \(R\), this motion has a constant acceleration \(g\) directed downwards (Earth's center).

Equations of vertical launch

The vertical launch is a uniformly varied motion, and considering the axes as shown in the figure above, the equations that will describe the movement are: $$ \left \{ \begin{array}{a} h(t) = h_0 + v_0 t - \frac{1}{2} g t^2\\ v(t) = v_0 - gt \\ v^2 = {v_0}^2 - 2 g \Delta h \end{array} \right . $$ where \(h(t)\) and \(v(t)\) is the height and the speed of the object in the time \(t\), \(h_0\) is the initial height, \(v_0\) initial speed and \(\Delta h\) is a height variation.

If the initial and final height are the same, then:

  • The time the object take to rise to the maximum height is equal to the time to descent to the initial height.
  • The speed modulus when the object is rising, at a specific height, is equal to speed's modulus when the object is descending.
The figure below illustrates the movement, first, an object is rising, latter on, the object is descending.