The Infinite Chase: Why Speed Took Centuries to Define

When we casually speak of speed, we imagine something straightforward: a car travelling at 60 miles per hour, a runner crossing the finish line, or a ball falling under gravity. Yet, as Feynman reminds us, the idea is deceptively subtle. Beneath the surface lies centuries of confusion, paradox, and ultimately the birth of an entirely new branch of mathematics: the calculus.

The Greeks, brilliant as they were in geometry and logic, struggled profoundly with motion. Their mathematical tools - ratios, areas, geometric constructions - were exquisitely suited to static problems. But motion is change in time, and time, slippery and continuous, resisted their grasp.

Consider the balloon problem that Feynman poses: if the volume of a balloon is increasing at a known rate, at what rate is the radius growing? To us, trained in calculus, this is an exercise in differentiation. To the Greeks, however, it was a riddle with no clear answer, for they lacked the conceptual machinery to express instantaneous change.

It was not for want of trying. Zeno of Elea, around the 5th century BCE, highlighted the very difficulties that would haunt thinkers for millennia. His paradoxes forced his contemporaries to confront the strange relationship between the finite and the infinite.

Take the most famous: Achilles and the tortoise. Achilles, swift as the wind, chases a tortoise granted a head start. By the time Achilles reaches the tortoise’s starting point, the creature has advanced a little further; when Achilles covers that, the tortoise has moved again; and so on, without end. Step by step, Achilles is forever behind.

The conclusion seems absurd, yet the reasoning is airtight—if one assumes that dividing time into an infinite series of steps means the task requires an infinite duration. What the Greeks could not see is that an infinite number of diminishing intervals can still sum to a finite time. Without the concept of a limit, the paradox remained a thorn in the side of ancient philosophy.

Fast forward two millennia, and subtleties remain. Feynman delights in recounting the tale of a woman stopped by a policeman for driving at 60 miles per hour. Her defence is both comical and profound: “How could I have been going 60 miles an hour when I wasn’t travelling for an hour?”

Here lies the modern echo of Zeno’s problem. The woman exploits ambiguity in our definition of speed. Are we speaking of an average rate over a fixed period, or of something more immediate, something happening at this very moment?

A policeman, naturally, need not quibble—“Tell it to the judge!” - but a physicist must. For the essence of speed, as we use it in physics, is not about travelling for an hour. It is about what happens in the tiniest slice of time imaginable.

To sharpen this idea, Feynman introduces the falling ball. At 5 seconds, how fast is it moving? One could calculate the distance travelled during the fifth second - 144 feet - and declare that the velocity is 144 feet per second. But this is an average. In reality, the ball is accelerating; at the start of that second it is slower, at the end faster.

To pin down the speed at exactly 5 seconds, one must look ever closer: the distance covered in a tenth of a second, a hundredth, a thousandth… dividing distance by smaller and smaller slices of time. The answer, he shows, stabilises at 160 feet per second.

This process - shrinking intervals towards zero and observing the limiting value - was utterly beyond the Greeks. It took Newton and Leibniz, independently in the 17th century, to formalise this as the differential calculus.

This is no longer an average speed stretched across an interval, but the instantaneous velocity, defined by a limit.

Having established what velocity means, Feynman shows how motion under constant acceleration naturally leads to one of physics’ most familiar formulas v-u=at.

If an object begins with velocity u and accelerates at rate a, then in a small time interval Δt its velocity changes by a Δt. Add these increments successively, and the velocity after time t becomes

v = u + at.

This simple, elegant expression - taught in every secondary school classroom - rests upon centuries of intellectual struggle. It encapsulates in a single line the outcome of the Greeks’ frustrations, Zeno’s paradoxes, a motorist’s evasive wit, and Newton’s revolutionary insight.

To say a car is moving at 60 miles per hour, or a stone at 160 feet per second, is to invoke one of the deepest achievements of human thought. We are not describing what happens across an hour or a second, but what is happening now, in an infinitesimal instant.

Feynman’s genius is to remind us that such a seemingly simple concept carries the weight of history. The Greeks could not define it, Zeno used it to confound his contemporaries, and even today it can be turned into a courtroom joke. Yet through calculus, motion has become something we can not only describe but predict, quantify, and harness.

In short, the expression v - u = at is not merely a formula. It is the resolution of ancient paradoxes, the clarification of ambiguous reasoning, and the crystallisation of Newton’s and Leibniz’s greatest insight: that motion lives in the infinitely small.