Speed, Velocity, and the Geometry of Motion

​In Chapter 9-2 of The Feynman Lectures on Physics, Richard Feynman pauses to sharpen a distinction that, in everyday language, we often blur: the difference between speed and velocity. Although the two words are used interchangeably in conversation, physics takes advantage of the linguistic redundancy to separate two subtly different ideas.

Velocity, in Feynman’s precise usage, is a vector: it possesses both a magnitude and a direction. Speed, by contrast, is reserved for the magnitude of that vector alone. In other words, speed tells us how fast something is moving, whereas velocity tells us how fast and in what direction.


Feynman illustrates this distinction by imagining an object moving through three-dimensional space. During a tiny interval of time, Δ t, the object experiences three small coordinate changes:


  • Δ x in the x-direction,
  • Δ y in the y-direction,
  • Δ z in the z-direction.



These increments form the edges of a small parallelepiped, and the actual motion of the object is a diagonal displacement Δ s across it. Each of the coordinate changes is simply the corresponding component of the velocity multiplied by Δ t. When combined through the geometry of three-dimensional space, these components produce the full displacement—a direct demonstration that velocity is something richer than speed.


What Feynman achieves here is more than a definition. He shows how differential geometry underpins even the simplest motions, and how vectors provide a language capable of capturing the intertwined nature of magnitude and direction. The distinction between speed and velocity is not pedantic; it is the foundation upon which much of Newtonian and modern physics is built.



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