# Oom Lecture 13

### Land Transport

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

$\Delta t_{heartbeat}\sim {\Delta V \over V}\propto {m_{heart} \over BMR}\propto m^{\frac {1}{4}}\,\!$ And in fact, the total number of heartbeats in a lifetime is independent of mass ($1.5\cdot 10^{9}$ ). This tells us that

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

$r\propto \ell ^{\frac {3}{2}}\,\!$ This gives us Then using that muscle strain goes as:

$\epsilon \sim {\Delta x \over \ell }\sim {2r\Delta y \over \ell ^{2}}\,\!$ And muscle stress is:

$\sigma \sim M_{young}\epsilon \sim \epsilon \sim E{2r\Delta y \over \ell ^{2}}\,\!$ We also use:

${f_{TG}\propto m^{-0.14}}\,\!$ where $TG$ is the trot-gallop transition.

We can also use:

{\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:

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