Linear Momentum, Impulse, and Collisions

Why do airbags save lives? They cannot change your momentum — that is fixed by how fast you were going. What they change is the time over which momentum drains away: $F\Delta t = \Delta p$, so ten times the stopping time means one tenth the force. Momentum thinking is collision thinking.

Linear momentum is defined as:

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

Newton's second law in terms of momentum: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt}$.

Impulse is the change in momentum produced by a force:

$$\vec{J} = \int_{t_i}^{t_f}\vec{F}\,dt = \Delta\vec{p}$$

Conservation of momentum — In a system with no external forces:

$$\vec{p}_{\text{total}} = m_1\vec{v}_1 + m_2\vec{v}_2 = \text{constant}$$

Collision types

Elastic collision in 1D (equal masses): the objects exchange velocities.

Centre of mass

$$\vec{r}_{\text{cm}} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2}{m_1+m_2}$$

The centre of mass moves as if all external forces act on a single particle of total mass $M$.

Key insight: Momentum conservation is a consequence of Newton's third law. Internal forces cancel in pairs, so only external forces can change the total momentum.

Common pitfall: Momentum is conserved in every collision, elastic or not — as long as no external force intervenes. What inelastic collisions lose is kinetic energy, never momentum. Conserve $p$ always; conserve $KE$ only when told the collision is elastic.