Why does kinetic energy increase quadratically, not linearly, with speed? (2011)
Kinetic energy’s quadratic growth with speed, expressed as ½mv², prompts both intuitive explanations and deeper questions about how physics is modeled and taught. Commenters explore multiple angles—work and power, potential energy, symmetry and relativity, and Lagrangian mechanics—while also highlighting how everyday examples like car braking and falling objects reveal the non‑linear stakes of going faster. A recurring theme is that physics often feels like a bag of “tricks” compared to the axiomatic clarity of math and computer science, raising questions about pedagogy, intuition, and what counts as a satisfying “why” in physical laws.
Core explanations of quadratic kinetic energy
- Several comments derive (E_k = \tfrac12 m v^2) from Newtonian basics:
- Start with (F = ma), work (W = F d), kinematics (v^2 = 2 a d).
- Combine to get (W = \tfrac12 m v^2), motivating why energy scales with (v^2), not (v).
- Others use differential form:
- Force is change of momentum over time; work is force over distance.
- Infinitesimal energy change at speed (v) is (dE = m v,dv); integrating yields (E \propto v^2).
- Connection to power: if power scales ∝ speed, integrating power over time naturally produces a quadratic dependence of energy on speed.
Intuitive and everyday analogies
- Car-braking anecdote: a faster car retaining large residual speed when both brake “the same” helps visualize how much extra energy higher speeds carry.
- Height/potential-energy analogy: doubling drop height doubles potential energy but less than doubles impact speed, showing energy vs speed can’t be linear.
- Everyday impacts (walking into a wall, axes vs mauls, hammers, car crashes) are used to make the “small speed increase → much larger damage” intuition vivid.
Nuances and counterexamples
- Multiple replies note that “same braking rate” is ambiguous: equal deceleration vs equal rate of energy dissipation. These lead to different intuitions.
- Real cars: downforce or lift makes deceleration rate speed-dependent, complicating the simple story but not the (v^2) law itself.
- One comment extends to relativity: the familiar quadratic term is just the low-speed part of a series; at high speeds kinetic energy grows faster than (v^2).
Alternative formulations and “what-if” universes
- Lagrangian/Hamiltonian viewpoints:
- The usual Lagrangian (L = \tfrac12 m v^2 - V(x)) plus symmetry requirements (Galilean invariance, homogeneity, isotropy) essentially forces a quadratic kinetic term.
- A thought experiment where energy were linear in speed shows it would break basic relativity and yield pathological dynamics.
- Some connect the quadratic form to dot products, rotation invariance, “spherical” geometry, and least-squares–type quantities.
Intuition gaps and pedagogy
- Several participants report physics feeling like a bag of tricks, unlike axiomatic math/CS, and struggle to build intuition even when they can do the math.
- Critiques of standard teaching:
- Overemphasis on formulas and “shut up and calculate”.
- Little historical context on how models were discovered.
- Post-hoc “obvious” derivations that hide the experimental basis.
- Suggestions include: classic textbooks, Lagrangian mechanics for mathematically minded readers, doing simple home experiments, and specialized texts on variational mechanics.
Deeper conceptual and meta questions
- Some ask whether energy and force are “real” or just bookkeeping; others note that which feels more fundamental depends on context (macroscopic vs quantum).
- There is debate over whether appeals to heat, temperature, and frame-dependent energy in some explanations are convincing or too hand-wavy.
- A side discussion examines StackExchange culture, long-term suspensions, and whether those communities are hostile or historically valuable, plus speculation about LLMs changing the landscape.