Momentum and Collisions

Quantity of Motion

Momentum measures an object's "quantity of motion", $p = mv$ — mass times velocity, a vector pointing the way the object moves. A heavy truck and a fast bullet can carry similar momentum for very different reasons. Momentum is one of the most powerful bookkeeping tools in physics because of its conservation law.

Impulse connects force and momentum: a force acting over a time changes momentum by $J = F\,\Delta t = \Delta p$. This is why a longer contact time softens an impact — an airbag or a bent-knee landing spreads the same momentum change over more time, reducing the peak force. Same $\Delta p$, smaller $F$ when $\Delta t$ is larger.

The headline result is conservation of momentum: in an isolated system (no external forces), the total momentum stays constant. In a collision between two objects, the momentum before equals the momentum after — even though the objects exchange velocities. This follows directly from Newton's third law (the equal-and-opposite internal forces cancel).

Collisions come in types by what happens to energy:

• Elastic — kinetic energy is also conserved (ideal, like billiard balls).
• Inelastic — some kinetic energy is lost to heat/deformation; in a perfectly inelastic collision the objects stick together.

The crucial point: momentum is conserved in all collisions (elastic and inelastic alike), whereas kinetic energy is conserved only in elastic ones. This distinction — momentum always conserved, kinetic energy only sometimes — is the key to analyzing any collision.

Common pitfall: assuming kinetic energy is conserved in every collision the way momentum is. Momentum is conserved in all collisions with no external forces; kinetic energy is conserved only in elastic collisions. Inelastic collisions (especially objects sticking together) lose kinetic energy to heat and deformation while still conserving momentum — treating them the same is the classic error.