Momentum becomes especially powerful when forces are large, complicated, and short-lived — collisions, explosions, catches, impacts.

Linear momentum is:

It is a vector.

Impulse changes momentum

Newton's second law can be written:

Integrating over time:

For an approximately constant force:

The area under a force-versus-time graph is therefore impulse.

Why airbags and crumple zones help

To stop a person, their momentum must change.

For the same Δp:

Increasing the stopping time reduces the average force.

That is the mechanical idea behind airbags, padding and vehicle crumple zones. They do not eliminate the required momentum change; they spread it over more time and often more distance.

Conservation of momentum

For a chosen system:

If external impulse is negligible:

Internal collision forces can be enormous and momentum can still be conserved, because the internal interaction forces exchange momentum within the system.

Collision types

Elastic collision

• momentum conserved
• kinetic energy conserved

Inelastic collision

• momentum conserved for an isolated system
• kinetic energy is not conserved as kinetic energy

Perfectly inelastic collision

• objects stick together afterward

Kinetic energy that disappears from the translational bookkeeping may become deformation, sound, thermal energy, vibration, rotation, or other internal energy.

Center of mass

For particles:

The center of mass of a system responds to net external force as though the total mass were concentrated there for translational motion:

This is why an exploding firework's fragments can fly in many directions while the system's center of mass continues along the trajectory determined by external forces.

Example: catching a ball

A 0.15 kg baseball moving at 40 m/s has momentum magnitude:

Stopping it in 0.010 s requires an average force magnitude around:

If the glove and arm move backward so stopping takes 0.050 s:

Same momentum change; five times the stopping time; one fifth the average force.

Physics C bridge

The force-time graph idea becomes exact through:

Calculus also makes variable-mass and continuous-system mechanics possible, although those are beyond the normal AP Physics C: Mechanics core.

Takeaway: Momentum is the natural bookkeeping quantity for interactions over time.