Main variables:
\( \vec{v} \) = Velocity.
\( \vec{a} \) = Acceleration.
\( t \) = Time.
\( \vec{F} \) = Force.
\( \vec{p} \) = Momentum.
\( \vec{I} \) = Impulse.
\( m \) = Mass.
\( K \) = Kinetic energy.
\( \hat{\textbf{i}} \) = Unitary vector along X-axis.
\( \hat{\textbf{j}} \) = Unitary vector along Y-axis.
\( \sum F\) = Sum over all the forces.
Subindex ‘i’ means initial. Subindex ‘f’ means final. For example, \(\vec{v}_i\) represents the initial velocity.
When we write \(a\) instead of \(\vec{a}\), we refer to the magnitude of that vector. For instance, \(\vec{a}\) is the acceleration vector, and \(a\) is the magnitude of the acceleration.
Main Equations:
Linear momentum:
\begin{equation*}
\vec{p} = m \vec{v}.
\end{equation*}
Relation between the force and the change of momentum:
\begin{equation*}
\sum \vec{F} = \frac{\Delta \vec{p}}{\Delta t}.
\end{equation*}
Newton’s second law in terms of the momentum:
\begin{equation*}
\sum \vec{F} = \frac{d \vec{p}}{dt} = m \vec{a}.
\end{equation*}
Impulse in terms of force:
\begin{equation*}
\vec{I} = \sum \vec{F} \Delta t.
\end{equation*}
Differential impulse in terms of force:
\begin{equation*}
d\vec{I} = \sum \vec{F} d t.
\end{equation*}
Impulse as a function of the momentum:
\begin{equation*}
\vec{I} = \Delta \vec{p} = \vec{p}_f – \vec{p}_i.
\end{equation*}
Conservation of momentum (when there are no external forces):
\begin{equation*}
\vec{p}_i = \vec{p}_f.
\end{equation*}
Conservation of momentum expanded for ‘n’ objects:
\begin{equation*}
m_1 \vec{v}_1 + m_2 \vec{v}_2 + \dots + m_n \vec{v}_n = m_1 \vec{v}_{f1} + m_2 \vec{v}_{f2} + \dots + m_n \vec{v}_{fn},
\end{equation*}
or
\begin{equation*}
\sum \vec{P}_i = \sum \vec{P}_f.
\end{equation*}
An elastic collision implies that the kinetic energy is conserved:
\begin{equation*}
K_i = K_f.
\end{equation*}
Elastic collision expanded for ‘n’ objects:
\begin{equation*}
\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 + \dots = \frac{1}{2} m_1 {v_f}_1^2 + \frac{1}{2} m_2 {v_f}_2^2 + \dots,
\end{equation*}
or
\begin{equation*}
\sum K_i = \sum K_f.
\end{equation*}
A completely inelastic collision for ‘n’ objects (the objects move together with the same velocity):
\begin{equation*}
m_1 \vec{v}_1 + m_2 \vec{v}_2 + \dots + m_n \vec{v}_n = \left(m_1 + m_2 + \dots + m_n \right) \vec{v}_f,
\end{equation*}
or
\begin{equation*}
\sum_k m_k \vec{v}_{ik} = \left( \sum_k m_k \right) \vec{v}_f.
\end{equation*}