Linear Momentum, Impulse, and Collisions

Momentum and Impulse

Airbags save lives by changing the time over which momentum drains away. Since $F\Delta t = \Delta p$, ten times the stopping time means one tenth the force.

Linear momentum is the product of mass and velocity: $\vec{p} = m\vec{v}$. Newton's second law states $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$.

Impulse is the change in momentum produced by a force over time: $\vec{J} = \int \vec{F}\,dt = \Delta\vec{p}$.

Conservation of momentum: When no external forces act, total momentum remains constant: $\vec{p}_{\text{total}} = m_1\vec{v}_1 + m_2\vec{v}_2 = \text{constant}$. This holds because internal forces cancel via Newton's third law.

Collisions and Centre of Mass

Momentum is conserved in every collision. What inelastic collisions lose is kinetic energy, never momentum.

For 1D elastic collisions with equal masses, objects exchange velocities.

The centre of mass position is $\vec{r}_{\text{cm}} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2}{m_1+m_2}$. It moves as if all external forces act on a single particle of total mass $M$.