Collisions

Two carts collide. During the crunch, the forces are huge, messy, and over in milliseconds — and you don't need any of them. Total momentum before equals total momentum after. That single line solves collisions.

The master equation

For any two objects colliding with no outside force:

$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$

(primes = after). Velocities carry signs — pick a positive direction and a head-on collision is just arithmetic with negatives.

The stick-together case

Simplest version — a moving cart hits an identical parked one and they couple:

Momentum m·v must now push twice the mass, so the pair rolls at v/2. Check the kinetic energy, though: it went from ½mv² to ½(2m)(v/2)² = ¼mv² — half the KE vanished into sound, heat, and bent metal. Momentum conserved, kinetic energy not.

Elastic vs inelastic

That distinction has names:

Momentum is in every row — it's conserved in all of them. Kinetic energy is the fragile one: any denting, heating, or noise spends it. A dropped ball that rebounds to 80% of its height just told you its bounce is inelastic.

One elastic result worth knowing by heart: equal masses colliding head-on-and-elastically swap velocities — the cue ball stops dead and the object ball leaves with its speed. Every pool player has run this experiment.

Explosions: collisions in reverse

Conservation doesn't care about the sign of time. A rifle fires: total momentum was zero, so bullet momentum forward = rifle momentum backward — that's recoil. Rocket exhaust, ice skaters pushing apart, a cannon rolling back: all the same one-liner, p_total = 0 throughout.

Why this matters — and where you've arrived

You now hold all three currencies of mechanics: forces for the play-by-play (F = ma), energy for totals without a clock, momentum for surviving collisions. Choosing the right one for the question in front of you — that's what "knowing mechanics" actually means. Take the arc you watched in the projectile simulator: forces built its equations, energy set its landing speed, and momentum says what happens when it hits.