Momentum

Definition 1: the momentum \(\mathbf{p}\) of a particle is defined as the product of the mass and velocity of the particle

\[ \mathbf{p} = m \mathbf{v}, \]

with \(m\) the mass of the particle and \(\mathbf{v}\) the velocity of the particle.

For the case that \(\mathbf{v}: t \to \mathbf{v}(t) \implies \mathbf{v}'(t) = \mathbf{a}(t)\) we have the following theorem.

Theorem 1: let \(\mathbf{v}\), \(\mathbf{a}\) be the velocity and acceleration of a particle respectively, if we have

\[ \mathbf{v}: t \to \mathbf{v}(t) \implies \forall t \in \mathbb{R}: \mathbf{v}'(t) = \mathbf{a}(t), \]

then

\[ \mathbf{p}'(t) = \mathbf{F}(t), \]

for all \(t \in \mathbb{R}\).

Proof:

Will be added later.