Linear Momentum

Notes

Momentum and impulse

• $\vec p = m\vec v$ and $\displaystyle \vec F_{net} = \frac{d\vec p}{dt}$.
• Impulse: $\displaystyle \vec J = \int \vec F\,dt = \Delta\vec p$. If $F(t)$ is given, integrate it.

Systems and the center of mass

• $\displaystyle \vec v_{cm} = \frac{\sum m_i\vec v_i}{\sum m_i} = \frac{\vec p_{total}}{M}$.
• With no net external force, the center of mass moves at constant velocity, even during an explosion.
• $\vec F_{ext} = M\vec a_{cm}$.

Collisions

• Momentum is conserved in every collision if external forces are negligible during the short impact.
• Elastic: kinetic energy is also conserved. In a 1D elastic collision, the relative speed of approach equals the relative speed of separation.
• Perfectly inelastic: objects stick: $\displaystyle v_f = \frac{m_1v_1 + m_2v_2}{m_1 + m_2}$.
• 2D collisions: conserve momentum separately in $x$ and in $y$.

Variable mass (rockets, sand dropping on a belt)

$\displaystyle F = \frac{dp}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt}$. A conveyor belt at constant speed $v$ collecting sand at rate $\displaystyle \frac{dm}{dt}$ needs extra force $\displaystyle v\frac{dm}{dt}$.