Conceptual Science

In earlier chapters of his Lectures on Physics, Richard Feynman explored motion in terms of distance and velocity. We learned that velocity is the rate at which distance changes - in other words, how fast and in what direction an object is moving. But Feynman now takes us a step deeper: instead of asking how far or how fast, he asks, how does the speed itself change over time? This is the idea of acceleration. From cars to falling bodies Feynman begins with a familiar comparison: a car that boasts about going from zero to sixty in ten seconds. That gives us a sense of average speed change, but what if we want to know the change at every instant? That’s what acceleration measures: how quickly velocity itself is shifting, second by second. To make this concrete, he turns to the simplest case - a freely falling object. We already know that, in such motion, velocity increases steadily as time goes on. By looking at how much the velocity grows each second, we discover that the increase is t...

In Chapter 8-4 of The Feynman Lectures on Physics, Feynman shifts focus from finding velocity given distance (differentiation) to the inverse problem: determining distance from a table of velocities. The key example is a falling ball, but he generalises to a car whose speed varies from moment to moment. Feynman emphasises that the question is not trivial. A constant speed gives an obvious distance. When speed varies, however, the problem becomes one of reconstructing distance from changing values. This requires a different way of thinking: not averages over long intervals, but the accumulation of small contributions. He demonstrates that dividing time into intervals Δt, as Δt becomes smaller, the approximation improves. The subtle point is his insistence that this procedure does not immediately give the exact distance. Accuracy depends on how small the intervals are, and the “limit” process is essential to move from approximation to precision. The integral symbol ∫ is introduced not as...

In Chapter 8-4 of  The Feynman Lectures on Physics , Richard Feynman introduces one of the central ideas of modern science and mathematics: the concept of  speed as a derivative . This chapter is not merely about mechanics; it is a gateway into calculus itself – the language that allows us to describe motion with precision. Feynman begins with a simple problem: how do we measure the speed of an object when its motion is not uniform? If something moves 100 metres in 10 seconds, its average speed is clear – 10 m/s. But what if the motion is irregular, changing from moment to moment? The average speed over a whole interval tells us little about the exact speed at a given instant. To resolve this, Feynman introduces the notation of increments: • Δt a small change in time, • Δs: the corresponding small change in distance. The ratio Δs / Δt is the average speed over a very small interval. As that interval becomes ever smaller, we approach the  instantaneous speed . Feynman th...

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 ex...