Main variables:

Main Equations:

The horizontal component of the velocity:

\begin{equation*}
v_x = |\vec{v}| \cos \theta.
\end{equation*}

The vertical component of the velocity:

\begin{equation*}
v_y = |\vec{v}| \sin \theta.
\end{equation*}

The speed (magnitude of the velocity):

\begin{equation*}
v = |\vec{v}| = \sqrt{ v_x^2 + v_y^2 }.
\end{equation*}

The angle (direction) of the velocity:

\begin{equation*}
\theta = \tan^{-1} \left( \frac{v_y}{v_x} \right).
\end{equation*}

The velocity of the horizontal component is constant:

\begin{equation*}
v_x = v_{ix} = v_{fx}.
\end{equation*}

Horizontal position as a motion with constant velocity:

\begin{equation*}
x_f = x_i + v_{x} t.
\end{equation*}

Vertical position as a motion with constant acceleration:

\begin{equation*}
y_f = y_i + v_{iy} t – \frac{1}{2} g t^2.
\end{equation*}

Velocity as a function of time:

\begin{equation*}
v_{fy} = v_{iy} – g t.
\end{equation*}

Velocity as a function of the position and the acceleration:

\begin{equation*}
v_{fy}^2 = v_{iy}^2 – 2 g ( y_f – y_i).
\end{equation*}

Motion with constant acceleration as a function of the velocities:

\begin{equation*}
y = y_0 + \frac{1}{2} (v_{iy} + v_{if}) t.
\end{equation*}

The maximum height can be obtained from:

\begin{equation*}
h_{\text{max}} = \frac{v_{i}^2 \sin^2 \theta}{2g}.
\end{equation*}

The maximum horizontal displacement is:

\begin{equation*}
x_{\text{max}} = \frac{v_0^2 \sin (2 \theta)}{g}.
\end{equation*}

Time to get to the maximum height:

\begin{equation*}
t = \frac{ v_y}{g} = \frac{ v_i \sin  \theta}{g}.
\end{equation*}

Time of flight:

\begin{equation*}
t = \frac{2 v_y}{g} = \frac{ 2 v_i \sin  \theta}{g}.
\end{equation*}

Trajectory:

\begin{equation*}
y = (\tan \theta) \, x  – \left( \frac{g}{2 v_i^2 \cos ^2 \theta } \right) x^2.
\end{equation*}