# Oom Lecture 13

### Land Transport

We can calculate the length of an animals heart-rate:

${\displaystyle \Delta t_{heartbeat}\sim {\Delta V \over V}\propto {m_{heart} \over BMR}\propto m^{\frac {1}{4}}\,\!}$

And in fact, the total number of heartbeats in a lifetime is independent of mass (${\displaystyle 1.5\cdot 10^{9}}$). This tells us that

${\displaystyle t_{life}\propto m^{\frac {1}{4}}\,\!}$

This is only for mammals.\We also have the elastic similarity (which says that the radius of the bone of a creature is related to the length of the creature):

${\displaystyle r\propto \ell ^{\frac {3}{2}}\,\!}$

This gives us Then using that muscle strain goes as:

${\displaystyle \epsilon \sim {\Delta x \over \ell }\sim {2r\Delta y \over \ell ^{2}}\,\!}$

And muscle stress is:

${\displaystyle \sigma \sim M_{young}\epsilon \sim \epsilon \sim E{2r\Delta y \over \ell ^{2}}\,\!}$

We also use:

${\displaystyle {f_{TG}\propto m^{-0.14}}\,\!}$

where ${\displaystyle TG}$ is the trot-gallop transition.

We can also use:

{\displaystyle {\begin{aligned}r&\propto m^{\frac {3}{8}}\\\ell &\propto m^{\frac {1}{4}}\\\end{aligned}}\,\!}

Assuming when we are running our feet are actually rolling without slipping over the ground, we have:

${\displaystyle v_{TG}\propto vf_{TG}\propto rm^{-{\frac {1}{8}}}\propto m^{\frac {1}{4}}\,\!}$