Why is “F = ma” still the most important equation in physics?
Why Read This
What Makes This Article Worth Your Time
Summary
What This Article Is About
Ethan Siegel explores why Newton’s second law (F = ma) remains physics’ most essential equation despite being 350 years old. The article demonstrates that this deceptively simple three-letter formula contains profound insights about motion, serves as the foundational application of calculus to physical reality, and provides pathways to understanding everything from differential equations to special relativity.
Beyond introductory physics, F = ma reveals itself as a second-order differential equation that enables predictions of future motion from current conditions, extends to angular momentum and rotating systems, and transforms into relativistic physics when properly understood as the rate of change of momentum rather than simply mass times acceleration. The equation’s enduring importance stems from its ability to connect simple observations about force and motion to sophisticated mathematical frameworks that describe nearly all classical physical phenomena.
Key Points
Main Takeaways
Physics Constrains Mathematics
F = ma cannot have a constant term because Newton’s first law requires objects to remain in motion without forces.
Three Equations in One
Force, mass, and acceleration operate independently in three spatial dimensions, explaining projectile motion and orbital mechanics.
Differential Equation Foundation
F = ma is a second-order differential equation relating acceleration to force, enabling predictions of future motion from current conditions.
Newton’s True Formulation
Newton actually wrote F = dp/dt, force as the rate of change of momentum, which accommodates relativistic effects and variable mass systems.
Extensions to Complex Systems
The equation extends naturally to angular motion, torque, extended objects, and systems of interacting particles through calculus-based methods.
Gateway to Relativity
When properly understood as momentum change, F = ma naturally incorporates relativistic effects including time dilation and length contraction.
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Article Analysis
Breaking Down the Elements
Main Idea
Enduring Relevance Through Mathematical Depth
The article’s central thesis demonstrates that Newton’s F = ma transcends its introductory physics origins to serve as the foundational differential equation of classical mechanics. Siegel argues that the equation’s continued importance across all physics education levels and professional applications stems from its mathematical sophistication as a second-order differential equation and its capacity to extend into relativistic physics when properly formulated as force equals the rate of change of momentum.
Purpose
Rehabilitating Undervalued Fundamentals
Siegel writes to challenge the physics community’s tendency to overlook F = ma in favor of more exotic equations, arguing that this pedagogical workhorse deserves recognition as physics’ most important equation. His purpose is both educational and advocative: to reveal hidden mathematical sophistication in apparently simple concepts while demonstrating how foundational equations contain pathways to advanced understanding. He aims to shift readers from viewing F = ma as merely introductory to recognizing it as eternally essential across all physics sophistication levels.
Structure
Progressive Complexity Layering
Expository Introduction → Mathematical Development → Advanced Extensions → Historical Revelation. Siegel begins by establishing the equation’s ubiquity and apparent simplicity, then systematically unpacks increasing layers of sophistication through connections to Newton’s three laws, three-dimensional vectorial nature, differential calculus, extended systems, and finally reveals that Newton’s original formulation naturally accommodates relativistic physics. The structure mirrors physics education itself, taking readers from high school understanding through graduate-level insights.
Tone
Enthusiastic, Pedagogical & Reverent
Siegel combines genuine enthusiasm for the equation’s elegance with clear pedagogical explanations accessible to multiple knowledge levels. His tone reveals deep respect for both Newton’s insight and the mathematical frameworks physics employs, while maintaining conversational accessibility through phrases like “let’s unpack” and direct addresses to readers. The writing balances technical precision with inclusive language that invites readers into increasingly sophisticated understanding without condescension.
Key Terms
Vocabulary from the Article
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Tough Words
Challenging Vocabulary
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Deviating from normal or expected behavior in a way that indicates fundamental problems or inconsistencies within a system or framework.
“any b that isn’t zero would lead to pathological behavior in physics”
Intended for or likely to be understood by only a small number of people with specialized knowledge or interest in a particular field.
“when we move on to rocket science, calculus, and some very intense, advanced, and esoteric concepts”
Describing systems where future states are completely and uniquely determined by current conditions and governing laws without any element of randomness.
“Newton’s equations are entirely deterministic, so if we can measure or know what an object’s initial conditions are at some time”
Relating to or exhibiting effects predicted by Einstein’s theory of relativity, particularly significant at speeds approaching the speed of light.
“so long as you remember to use relativistic momentum (where you add in the Lorentz factor)”
An object or person that enters a place or situation where it does not belong or is not expected to be.
“It’s even how we determined the orbit of our newest interstellar interloper: 3I/ATLAS”
A substance burned or expelled to provide thrust and forward motion to rockets, missiles, or other vehicles through the expulsion of mass.
“calculating the behavior of a spacecraft powered by propellant would be impossible”
Reading Comprehension
Test Your Understanding
5 questions covering different RC question types
1According to the article, F = ma can be written as F = ma + b where b represents a constant force, similar to how y = mx + b represents a line in mathematics.
2What is the primary reason the article gives for why F = ma is called a “second-order” differential equation?
3Which sentence best explains why what happens in one spatial dimension does not affect other dimensions in F = ma?
4Evaluate whether each statement about Newton’s formulation of his second law is true or false.
Newton originally wrote his second law as force equals the time rate of change of momentum, not as F = ma.
Newton’s original formulation fails to work for systems where mass changes over time, such as rockets expelling fuel.
When properly formulated with relativistic momentum, Newton’s force law naturally produces the effects of special relativity including time dilation.
Select True or False for all three statements, then click “Check Answers”
5Based on the article’s discussion of differential equations and F = ma, what can be inferred about the relationship between mathematical sophistication and the perceived importance of physical equations?
FAQ
Frequently Asked Questions
A differential equation relates a quantity to its rate of change. Since acceleration is the rate at which velocity changes (and velocity is the rate at which position changes), F = ma is a second-order differential equation that tells you how motion evolves over time. Given an object’s current position, velocity, and the forces acting on it, this differential equation allows you to predict exactly where that object will be at any future moment, making it the mathematical foundation for understanding all classical motion.
Newton’s original formulation as “force equals the rate of change of momentum” is more general than F = ma because momentum (p = mv) can change either through changing velocity or changing mass. This formulation naturally handles rockets that lose mass as they burn fuel and, remarkably, when combined with relativistic momentum definitions, it produces all the effects of special relativity including time dilation and length contraction. Some speculate Newton anticipated relativity’s insights, though it’s equally plausible he simply recognized that mass itself might vary in physical systems.
F = ma represents the most important application of calculus to physical reality. Acceleration is the derivative of velocity with respect to time (a = dv/dt), and velocity is itself the derivative of position with respect to time (v = dx/dt). This creates a chain of derivatives connecting force to position through two levels of time rates of change. Understanding F = ma deeply requires mastering how derivatives describe instantaneous rates of change and how differential equations predict future states from present conditions—concepts that form the mathematical language of all physics.
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This article is rated Advanced because it requires understanding sophisticated mathematical concepts including differential equations, calculus derivatives, vector mathematics, and relativistic physics. The vocabulary includes technical terms like “pathological behavior,” “second-order differential equation,” “Lorentz factor,” and “deterministic systems.” While Siegel makes these concepts accessible through clear explanations and examples, full comprehension demands comfort with abstract mathematical reasoning and the ability to follow multi-layered conceptual arguments about how simple equations contain profound mathematical depth.
Siegel highlights this point to underscore both the difficulty and value of F = ma: even though it’s a challenging second-order differential equation, physicists have developed methods to solve it in many important cases. This makes the “simple” F = ma extraordinarily powerful—it’s mathematically sophisticated enough to be genuinely difficult, yet tractable enough to yield exact solutions for planetary motion, projectile trajectories, and countless other phenomena. The fact that we can solve it (unlike most differential equations) while it remains genuinely challenging represents an ideal balance between mathematical depth and practical applicability.
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