MOMENTUM AND COLLISIONS

A cue ball crosses the felt and strikes a resting ball dead centre. The cue ball stops. The other ball shoots forward at exactly the speed the cue ball had a moment before. As if a quantity of motion had been handed off, whole and intact, from one body to the next.

That quantity has a name. It is called momentum, usually written p, and it is the subject of this topic.

Energy alone cannot tell you how that collision will end. Two equal balls approach each other at 3 m/s each. After they collide, they might both stop, swap velocities, fly apart at some other pair of speeds — kinetic energy is the same in any of those outcomes (if the collision is elastic). Energy narrows the possibilities. It does not pin down the answer.

Momentum does. The total momentum before — m·(+3) + m·(−3) = 0 — must equal the total momentum after. Whatever the final velocities are, they have to come in equal and opposite pairs. Combined with energy conservation, that forces a unique outcome: they swap. Every collision problem, from pool hall to rocket launch, is both ledgers applied at once.

Newton wrote his second law not as F = ma but as F = dp/dt. The form matters. Force is the rate at which momentum changes. It generalises cleanly into relativity. It survives a rocket losing mass mid-flight. F = ma is a special case. F = dp/dt is the real thing.

Momentum is the product of mass and velocity:

A vector — magnitude and direction, both. A 1 kg object moving at 3 m/s east has momentum +3 kg·m/s east. The same object moving west has −3. In three dimensions it's a triple (p_x, p_y, p_z), one conservation law per axis.

The law itself is one line:

for any closed system — one no external force is acting on. Internal forces between the parts don't matter. If particle A pushes particle B, Newton's third law says B pushes back on A with an equal and opposite force; the total p_A + p_B doesn't budge. Conservation of momentum is Newton's third law, promoted from a statement about forces to a statement about their time-integral.

Descartes had the idea first, in 1644, and called it quantitas motus — quantity of motion. He thought it was the deepest law of nature. He was mostly right. What he got wrong was the definition: he used |m·v|, absolute value, no sign. That version doesn't conserve. Within a generation Huygens and John Wallis, working independently in 1668, caught the error, added the sign, and made the law actually hold — Huygens on elastic collisions, Wallis on inelastic ones. Newton wrote it into the Principia a generation later, where it has lived ever since.

If a force F acts on a body for a time Δt, it delivers an impulse J = F·Δt. By F = dp/dt, the impulse is exactly the change in momentum:

For a varying force, J = ∫F·dt, with the integral taken over the duration of the force. The insight is that it is the integral of force over time that matters for changing momentum, not the peak force — and this fact underpins every piece of safety engineering there is.

Airbags are the classic illustration. Stop a 70 kg passenger travelling at 20 m/s and you must remove 1400 kg·m/s of momentum. That total is fixed. What you can control is how quickly. Hit a steel dashboard and it happens in maybe 5 ms — average force 280 kN, roughly 400g, bones shatter. Hit an airbag and it happens in maybe 100 ms — average force 14 kN, survivable. Same impulse, different durations, spectacularly different peak forces. Every crumple zone, foam helmet lining, parachute, and landing-gear shock absorber does the same trick: trade a long Δt for a low F, at the expense of moving distance.

Same logic, opposite direction: a karate chop, a golf drive, or the crack of a bullwhip wants a very short Δt — hit hard and fast — to produce the biggest possible force from a limited impulse budget. Catching an egg with both hands gently pulled back mid-catch gives you a long Δt and keeps the egg intact. Catching it against a wall gives you a short Δt and a broken yolk.

A system of several particles has a single special point called the center of mass, defined by

which in words is the mass-weighted average position of all the parts. Differentiate both sides with respect to time and you get

where M is the total mass. The total momentum of a system is exactly the total mass times the velocity of its center of mass. If external forces sum to zero, total momentum is conserved, and therefore the center of mass moves in a straight line at constant velocity — forever.

This is a startlingly powerful statement. Consider a hand grenade thrown in an arc, which explodes mid-flight into a hundred fragments flying in every direction. Every single fragment has its own messy trajectory. But the center of mass of all those fragments, summed, continues along the original parabola as if the explosion never happened. The internal forces of the explosion cancel in pairs — they can redistribute momentum among fragments but cannot move the center of mass. Ignore air resistance, and the center of the swarm lands exactly where the unexploded grenade would have landed.

Divers use the same trick. A tuck-somersault diver spins rapidly in mid-air, but their center of mass follows a perfect parabola from the springboard to the water. The limbs trace complicated curves; the CM does not. Trampolinists, gymnasts, and astronauts all rely on the same separation: internal rearrangements change individual motion but cannot shift the system's center of mass.

Two unequal masses, rigidly linked, tumble as they translate. Each mass traces a complicated looping path. The magenta dot marking their common CM glides along in a perfectly straight line. Whatever the bodies do between themselves, the CM keeps sailing at constant velocity — that is the translational motion of the system, cleanly decoupled from everything internal.

Two objects meet, exchange some force for a short Δt, and go their separate ways. Between the first contact and the last contact, no external forces act (we can ignore gravity and friction during the short collision). So momentum is conserved. That much is always true.

What's not always true is whether kinetic energy is conserved. Three regimes:

A collision is elastic if kinetic energy is conserved. Billiard balls come close. Hard steel spheres come closer. Protons scattering off other protons are essentially perfect. Rubber bouncy balls on a hard floor can retain 90% of their KE per bounce.

A collision is perfectly inelastic if the maximum possible KE is lost — the two bodies stick together and move as one. Cars in a pile-up, a lump of putty on a block, an arrow in a target. Momentum is still conserved, but KE drops by a factor set by the mass ratio. Some of it has gone into heat, sound, and deformation of the crumpled metal.

In between, most real collisions are partially inelastic. They're parameterised by a coefficient of restitution e, defined as the ratio of post-collision to pre-collision relative speed. e = 1 is elastic, e = 0 is perfectly inelastic, and everything real lies between. A tennis ball on a hard court has e ≈ 0.75; a baseball on a wooden bat has e ≈ 0.55; a dropped marble on concrete has e ≈ 0.9.

Slide e down to zero and the two bodies merge — KE drops by the factor m_B / (m_A + m_B) relative to the initial KE, which is the fraction always lost in a perfectly inelastic collision. Slide mass ratio up and watch A bounce backwards when it hits a much heavier B — the classic reflection, the same mechanism that makes a tennis ball shoot up faster than the incoming racket. The running total at the top stays fixed: momentum is constant through the entire process, regardless of e, regardless of mass ratio. Only the kinetic-energy bar shrinks.

The math for the 1-D case is worth seeing. With A moving at v_A and B at rest, an elastic collision gives:

Three limits are worth noticing. When m_A = m_B, the first ball stops dead and the second takes its velocity — the Newton's cradle trick. When m_A ≫ m_B, A barely slows and B shoots off at 2·v_A — this is why a golf club (heavy) sends a golf ball (light) at roughly twice the clubhead speed. When m_A ≪ m_B, A bounces back at almost −v_A and B hardly moves — which is why a rubber ball bounces off a wall.

In a closed system you cannot change the total momentum. You can, however, rearrange it — expel some mass backward, and the rest moves forward. That is how rockets work.

A rocket of instantaneous mass M expels fuel at exhaust velocity u (relative to itself) at a mass flow rate dm/dt. Momentum balance: the fuel that leaves carries momentum (dm)·(−u) backward; the rocket, for its total momentum to stay fixed, must gain momentum M·dv forward. Over an infinitesimal interval,

Integrate from initial mass M₀ to final mass M:

That logarithm is the cruellest function in engineering.

Konstantin Tsiolkovsky derived it in 1903 — a deaf schoolteacher in the Russian provincial town of Kaluga, with no laboratory, no colleagues, and almost no readers outside a small circle of Moscow enthusiasts. He worked it out from nothing but Newton's laws and the willingness to track what happens when mass itself is flowing out of the object under study. The rocket equation is what fell out. A hundred and twenty years later it is still what NASA mission planners solve first when they sketch any trajectory, from LEO to Pluto.

The logarithm is the villain. Every kilogram of payload you want to move requires fuel; the fuel for that fuel; the fuel for that fuel's fuel; all exponentially. Reaching low Earth orbit demands about 9.4 km/s of Δv. Chemical rockets have exhaust velocities around 4.5 km/s. So the required mass ratio is M₀/M ≈ e^(9.4/4.5) ≈ 8. Eight kilograms of fuel per kilogram of rocket-plus-payload sitting on the pad. That is why a Saturn V was 85% propellant by mass, and why spacecraft are engineered to shed every empty tank they can.

The logic also runs in reverse. If a squid wants to move forward, it squirts water backward. If a human on a frictionless ice rink wants to move, they throw a shoe. Momentum conservation is the reason no body can propel itself forward without pushing something else backward — there is no magic thruster, and there never will be.

We now have two conservation laws: energy (from FIG.05) and momentum (from this topic). They do not conflict — they are independent constraints — and applied together they pin down almost any collision in classical mechanics. Every billiard game, every car crash analysis, every rocket trajectory, and every bouncing ball is a balance of these two ledgers.

One more conservation law is coming in FIG.07. It is the rotational cousin of what we did here — mass times velocity becomes moment-of-inertia times angular velocity, and the same argument that pinned momentum to translational symmetry will pin a new quantity, angular momentum, to rotational symmetry. Skaters pulling in their arms, hurricanes tightening over warming oceans, pulsars spinning thousands of times a second after a star's death — all of these are angular-momentum conservation in action. The pattern keeps repeating. By FIG.08 we will finally see why.