Components of Velocity, Acceleration, and Force A Gentle Walk Through the Geometry of Motion

​One of the joys of reading Feynman is discovering how something that initially feels complicated becomes strikingly simple once seen from the right angle. Chapter 9-3 of The Feynman Lectures on Physics is a perfect example. Here, Feynman takes what seems like an intimidating idea - motion changing in three dimensions under the influence of forces - and reveals a clear geometric structure beneath it.

Imagine watching an object move through space. It darts forwards, rises, drifts sideways - three motions woven into one. Instead of wrestling with the movement as a whole, Feynman suggests we split it into three threads: the motion along the horizontal axis, the vertical axis, and the depth axis.

This approach is powerful because each thread can be described independently. If you know how fast the object is moving horizontally, how fast it’s rising or falling, and how fast it’s shifting in depth, then you know everything about its velocity - both how fast it is going and the direction in which it is travelling.

The overall “speed” is simply the geometric combination of these three directional rates. But the direction-by-direction description is the one that unlocks the physics.

Now imagine a force acts on the object. Perhaps it nudges it sideways, pulls it downwards, or swings it into a new trajectory. Forces change velocities, and they do so in a way that can again be split into components.

The change in velocity in each direction - say, the change in horizontal motion over a blink of time - tells you the “acceleration” in that direction. This is the key idea: acceleration also has components, just like velocity.

Feynman emphasises that Newton’s Second Law, often written as a single relationship between total force and total acceleration, is really three separate statements operating side by side:

• The horizontal component of the force controls the horizontal acceleration.

• The vertical component controls the vertical acceleration.

• The depth component controls the depth acceleration.

In other words, nature handles each direction independently, unless forces couple them together.

Just as velocity and acceleration can be projected onto the three axes, forces can too. Any force - no matter its direction - can be imagined as a combination of three simpler pushes: one purely horizontal, one purely vertical, and one purely in or out of the page.

Once you know these components, Newton’s laws tell you exactly how each part of the motion will change.

This way of thinking turns multidimensional dynamics into something more like accounting: each direction gets its own ledger, its own credits and debits, its own running total of motion.

Feynman recalls a demonstration from an earlier chapter: an object is launched horizontally while it simultaneously falls under gravity. The intriguing result is that its horizontal motion continues untouched - smooth, steady, unchanging - while its vertical motion behaves exactly as it would if the object were simply dropped.

Gravity influences only the vertical component, not the horizontal. The two motions coexist but do not interfere.

This is one of the most striking lessons in early mechanics: motions in different directions remain independent unless a force links them.

The component approach is not just a mathematical trick; it is the conceptual backbone of mechanics. From planetary orbits to the flight of a ball, from designing bridges to predicting rocket trajectories, engineers and physicists rely on this decomposition. It allows them to treat complex problems as collections of simpler ones.

Feynman’s gift here is reminding us that the splendour of three-dimensional motion does not require three-dimensional confusion. Break the motion apart, understand each part on its own terms, and the whole picture becomes transparent.