Classical Mechanics · Momentum & Collisions
Every explosion, collision, and rocket burn obeys a single principle: the total momentum of an isolated system never changes. This conservation law — more fundamental than Newton's Laws — is true even in quantum mechanics and relativity.
Conservation of momentum is one of the most fundamental laws in all of physics. It is more fundamental than Newton's Laws — it holds even in quantum mechanics, even in special relativity, even in situations Newton never imagined. The universe simply will not allow total momentum to change without an external force.
Idea 1 · What is Momentum?
Momentum (p) is a vector — it has both magnitude and direction. A massive, slow object can have the same momentum as a light, fast one. That's why a slow freight train is harder to stop than a fast bicycle, even if the bicycle is moving quicker.
F = ma is actually a special case. The full law is: F = dp/dt — force equals the rate of change of momentum. This version works even when mass changes (like a rocket burning fuel).
In an isolated system (no external forces), total momentum is constant:
"Isolated" means no outside forces act. For collisions that last milliseconds, internal forces are enormous compared to gravity or friction, so we treat them as isolated.
By Noether's theorem: conservation of momentum follows from the fact that the laws of physics are the same at every point in space (translation symmetry). If the universe looks the same here as 1 meter away, momentum must be conserved.
Idea 2 · Newton's Third Law → Momentum Conservation
Newton's Third Law says: for every action force, there is an equal and opposite reaction force. These forces act on different objects — they never cancel each other on the same object.
But when you apply this to two objects interacting: the force on object 1 from object 2 is equal and opposite to the force on object 2 from object 1. Since Δp = FΔt:
The center of mass of an isolated system moves at constant velocity — forever. No internal forces (collisions, explosions) can change it. Only external forces shift the center of mass trajectory.
Idea 3 · Impulse — Force Over Time
Impulse is force multiplied by the time it acts. It equals the change in momentum. This is why extending the contact time reduces the peak force — the same impulse (same Δp) spread over more time means less force each instant.
On a force vs. time graph, the impulse is the area under the curve. A short, sharp spike of force and a long, gentle push can deliver the same impulse (same area) but completely different maximum forces.
A baseball bat swung faster stays in contact with the ball for a slightly shorter time — but the force is so much larger that the impulse (and therefore the ball's change in momentum) is greater. Follow-through extends contact time, increasing total impulse.
Idea 4 · Relativistic Momentum
Newtonian momentum p = mv breaks down near the speed of light. Einstein's relativity requires a correction factor γ (gamma). As v → c, γ → ∞ — meaning it takes infinite force to accelerate an object to the speed of light.
At everyday speeds, γ ≈ 1, so p ≈ mv — classical momentum is fine. But particle accelerators push protons to 99.9999% of c, where γ ≈ 7,000 and the relativistic correction is enormous.
The full energy-momentum relation. When p = 0 (object at rest), E = mc² — the famous rest energy. When m = 0 (a photon), E = pc — light carries momentum even with zero mass.
Even though photons have zero rest mass, they carry momentum p = E/c = hf/c (where h is Planck's constant and f is frequency). Solar sails use this — sunlight pushes on a large reflective sail to propel a spacecraft without fuel.