In 1686, Gottfried Wilhelm Leibniz proposed that the true measure of a body's "force of motion" is mv² — mass times velocity squared. He called it vis viva, living force. He derived it from Galileo's falling bodies, where v² = 2gh, by multiplying both sides by mass. Modern physics inherited this as kinetic energy: ½mv².
The quantity has been treated as fundamental for over three centuries. But it has a property that should disqualify it from describing physical reality: it is frame-dependent. The kinetic energy of an object changes depending on who is observing it. Not by a little. Not in a subtle way. It changes completely.
Two galaxies approach each other at relative velocity v. From galaxy A's frame, galaxy B carries all the kinetic energy. From galaxy B's frame, galaxy A carries all of it. From a frame where both move at v/2, the total is different from either. The "energy in the system" — supposedly a real, conserved, physical quantity — gives a different number depending on where you stand.
Momentum's value also changes between frames. But momentum has a crucial advantage: its conservation law — that total momentum before equals total momentum after a collision — holds identically in every inertial frame. And there is a quantity that is even more honest: the relative velocity between two objects. Its value doesn't change at all. It is the same number for every observer, everywhere, without qualification.
Two objects approach each other. Drag the observer frame slider and watch all three quantities: kinetic energy (fluctuates wildly), momentum (value changes but conservation law holds), and relative velocity (never changes at all).
An elastic collision between two objects. The left panel solves it using momentum + the relative velocity condition. The right panel solves it using momentum + energy conservation. The answers are always identical. Energy adds nothing.
The velocity-squaring operation does two things that should concern anyone who cares about physical reality.
First, it destroys direction. Velocity is a vector — it points somewhere. The moment you square it, a ball moving left and a ball moving right at the same speed have identical "kinetic energy." The one piece of physical information that distinguishes them — direction — is erased.
Second, it creates frame-dependence. Because squaring is nonlinear, boosting to a different frame doesn't just shift the energy — it changes its structure. The total kinetic energy of a system is not invariant under change of reference frame. This is not a minor technicality. It means the quantity has no unique physical value. It is not a property of the objects. It is a property of the observer's arbitrary choice of coordinates.
Momentum's value also changes between frames — as the demo above shows honestly. But momentum has something energy doesn't: its conservation law holds identically in every frame. If momentum is conserved in one frame, it's conserved in all of them. Energy conservation, by contrast, gives different totals in different frames, making it impossible to assign a unique "energy of the system" without first choosing a privileged frame.
But the cleanest quantity of all is the relative velocity — the speed of approach or separation between two objects. It doesn't just have a frame-independent conservation law. Its value is the same in every frame, for every observer, without exception. And as the collision solver demonstrates, momentum plus the relative velocity condition is sufficient to predict every elastic collision outcome without ever invoking energy.
The hierarchy is clear. Relative velocity: frame-independent value. Momentum: frame-dependent value, frame-independent conservation law. Kinetic energy: frame-dependent value, frame-dependent total — the weakest of the three, and the one Leibniz placed at the foundation.