¶Momentum, Impulse, and Conservation Laws

Linear momentum is a fundamental conserved quantity in classical mechanics. For a particle of mass $m$ moving with velocity $\vec{v}$:

$\vec{p} = m\vec{v}$

¶Newton's Second Law (Momentum Form)

Newton's original formulation of his second law: $\vec{F} = \frac{d\vec{p}}{dt}$

For constant mass, this reduces to $F = ma$. For variable mass systems (rockets burning fuel): $F = m\frac{dv}{dt} + v_e\frac{dm}{dt}$ where $v_e$ is the exhaust velocity.

¶Conservation of Linear Momentum

In an isolated system (no net external force): $\vec{p}_{\text{total}} = \sum_i m_i \vec{v}_i = \text{constant}$

¶Types of Collisions

In a perfectly inelastic collision, objects stick together: $m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f$

¶Impulse–Momentum Theorem

$J = \int_{t_1}^{t_2} F\,dt = \Delta p$

For constant force: $J = F \Delta t$.

¶The Tsiolkovsky Rocket Equation

From momentum conservation for variable mass systems: $\Delta v = v_e \ln\left(\frac{m_0}{m_f}\right)$

¶References

• Classical Mechanics (John R. Taylor, University Science Books)
• Feynman Lectures, Vol. I, Ch. 10: Conservation of Momentum
• NASA: How Rockets Work