#### Mass Comes from Volume Mass is essentially **density × volume**: \[ M = \rho V \] where: - \(M\) = mass - \(\rho\) = density - \(V\) = volume For most organisms, density is roughly constant (close to the density of water). Therefore: \[ M \propto V \] If we model organisms as having a characteristic body length \(L\), then: \[ V \sim L^3 \] So mass scales as: \[ M \sim L^3 \] This implies: \[ L \sim M^{1/3} \] --- #### Rewrite Surface Area in Terms of Mass Surface area scales with the square of length: \[ A \sim L^2 \] Substitute the relation \(L \sim M^{1/3}\): \[ A \sim (M^{1/3})^2 \] \[ A \sim M^{2/3} \] Muscles produce force by contracting fibers. Imagine a muscle shaped like a cylinder. The force it can produce depends on its cross-sectional area, not its length. But force per area is nearly constant across animals, because all muscles use the same molecular machinery. Galileo noticed that: - **Strength ∝ cross-sectional area** - **Weight ∝ volume** Mathematically: \[ \text{Strength} \propto L^2 \] \[ \text{Weight} \propto L^3 \] where \(L\) is the characteristic body length. Because weight increases faster than strength as animals get larger, **larger animals are relatively weaker** (in terms of strength relative to their body weight). This is why large animals must compensate structurally — for example: - **Elephants have very thick legs** - **Large animals have more column-like limb structures** - **Small animals can appear much stronger relative to their size** One beautiful consequence is that the paper predicts a common locomotor clock of about 0.1 s per body length at max speed. The authors later extended this idea: their 2021 paper on response times argued that across organisms from bacteria to large vertebrates, most simple stimulus-response times cluster around 25 ms, only a few times smaller than this locomotor timescale. That is a lovely hint that biology may operate with a small set of broadly shared dynamical timescales. A beautiful multi-scale connection: The scaling depends ultimately on $k_BT$ via the molecular motor energy, so the speed of animals is ultimately limited by thermal physics at the nanometer scale! ## TL;DR Because density, muscle stress, and metabolic power per mass are nearly universal across life, organisms (from bacteria to mammals) naturally end up with a maximum speed of roughly 10 body lengths per second. \[ \frac{V_{\max}}{L} \approx 10 \ \text{s}^{-1} \] Meaning: most organisms can move at about **10 body lengths per second**, despite **~20 orders of magnitude variation in mass**. The authors argue that this universal scaling comes from three biological quantities that are roughly constant across life: 1. **Body density** \[ \rho \approx 10^3 \text{ kg/m}^3 \] ≈ density of water. 2. **Muscle stress (force per cross-section)** \[ \sigma \approx 2\times10^5 \text{ N/m}^2 \] Set by molecular motors like myosin. 3. **Maximum metabolic power per mass** \[ b_M \approx 2\times10^3 \text{ W/kg} \] Using dimensional reasoning and energy limits on locomotion cycles, they derive: \[ \frac{V_{\max}}{L} \sim \frac{b_M \rho}{\sigma} \] Plugging in numbers gives: \[ \frac{V_{\max}}{L} \approx 10 \text{ s}^{-1} \] which matches observations. #### Why Large Animals Break the Law For large animals another constraint appears: **inertia of limbs**. #### Torque applied \[ C = \sigma S L \] #### Moment of inertia \[ I = M L^2 \] Since \[ M = \rho S L \] then \[ I = \rho S L^3 \] ### Angular acceleration \[ \alpha = \frac{C}{I} \] Substituting: \[ \alpha = \frac{\sigma}{\rho L^2} \] Thus **limb angular acceleration decreases with body size**. #### Consequence This leads to a **maximum absolute running speed** \[ V_{max} \sim \sqrt{\frac{\sigma}{\rho}} \] which becomes **independent of body size**. #### This explains why animals like - cheetahs - horses - ostriches all reach similar **top speeds around 20–30 m/s** despite large differences in body mass.