DLangrish

$mg\sin \theta =\mu mg\cos \theta +{\frac {\rho AC_{d}v^{2}}{2}}$

Where:

• m = mass
• g = gravitational acceleration constant for earth
• sin(theta) = factor of weight acting down the slope
• mu = co-efficient of friction
• cos(theta) = factor of weight acting perpendicular to the slope
• rho = air density
• A = cross-sectional area
• Cd = coefficient of drag
• v = velocity

Re-arranged in terms of v:
$v={\sqrt {\frac {2(mg\sin \theta -\mu mg\cos \theta )}{\rho AC_{d}}}}$

Example to estimate terminal velocity for me in a speed tuck on a 10% slope. $v={\sqrt {\frac {2(83*9.81\sin(5.71)-0.022*83*9.81\cos(5.71))}{1.2*0.37*1}}}=16.87ms^{-}1$

That's 37.4mph. Not unrealistic I would hope.

$mg\sin \theta =\mu mg\cos \theta +{\frac {\rho AC_{d}v^{2}}{2}}$

$k={\frac {2(\rho _{s}-\rho _{m})g}{9\eta }}$

$\eta ={\frac {2(\rho _{s}-\rho _{m})g}{9k}}$ $v_{t}=kr^{2}$

$F=6\pi r\eta v$

$C={\frac {2F}{\rho v^{2}A}}$

$m=m_{added}.{\frac {1}{({\frac {f_{1}}{f_{2}}})^{2}-1}}$

$m{\ddot {x}}+c{\dot {x}}+kx=Fe^{j\omega t}$

$Z={\frac {F}{\dot {x}}}$

${\dot {x}}={\dot {x}}_{0}e^{j\omega t}$

$Z={\frac {F}{\dot {x}}}{\frac {e^{j\omega t}}{e^{j\omega t}}}=j\omega m+c-{\frac {jk}{\omega }}$

$Re(Z)=c$

$Im({\tilde {Z}})=\omega m-{\frac {k}{\omega }}$

$\delta ={\frac {1}{n}}ln{\frac {x_{1}}{x_{n}+1}}$

$\zeta ={\frac {\delta }{\sqrt {(2\pi )^{2}+\delta ^{2}}}}$

$\eta =2\zeta$

$\delta =ln{\frac {x_{1}}{x_{2}}}$

$\zeta ={\frac {\omega _{2}-\omega _{1}}{2\omega _{n}}}$

${\tilde {H}}$

${\tilde {R}}$

$u={\sqrt {\frac {2(p_{T}-p)}{\rho }}}$

${\tilde {Z}}$

$f_{c}={\frac {1.84c}{\pi d}}$

$q=Au$

$Power={\frac {1}{2}}\rho u^{2}q$

$\Delta {\dot {E}}_{mech}\approx (p_{9}-p_{8})q$

$({\frac {P}{P_{0}}})^{\frac {1}{\gamma }}={\frac {\rho }{\rho _{0}}}=({\frac {T}{T_{0}}})^{\frac {1}{(\gamma -1)}}$

$P=e^{\frac {20.386-5132}{T}}$

$\omega ={\sqrt {\frac {2k}{m_{2}}}}$

$P_{total}$

$p_{i}$

$P_{total}=p_{air}+p_{w}$

${\frac {P_{total}}{P_{w}}}={\frac {p_{air}+p_{w}}{p_{w}}}={\frac {p_{air}}{p_{w}}}+1$

$pV=nRT,p={\frac {nRT}{V}}$

${\frac {p_{air}}{p_{w}}}$

${\frac {p_{air}}{p_{w}}}={\frac {({\frac {n_{air}RT}{V}})}{({\frac {n_{w}RT}{V}})}}={\frac {n_{air}}{n_{w}}}$

$h=u+p{\frac {1}{\rho }}$

$dh=du+dp{\frac {1}{\rho }}-p{\frac {dp}{\rho ^{2}}}$

${\frac {d\rho }{\rho ^{2}}}$

$dh={\frac {dp}{\rho }}$

${\frac {dh}{dz}}={\frac {1}{\rho }}{\frac {dp}{dz}}$

$p(z)=p_{0}-z\rho g$

${\frac {dp}{dz}}=-\rho g$

${\frac {dh}{dz}}=-g$

$c_{p}{\frac {dT}{dz}}=-g$

${\frac {dT}{dz}}={\frac {-g}{c_{p}}}$

$-\int {\frac {dT}{dz}}dz=\int {\frac {g}{c_{p}}}dz$

$-T(z)={\frac {g}{c_{p}}}z-constant$

${\frac {constant}{c_{p}}}=konstant=T(z)c_{p}+gz$

$konstant=h(z)+gz$

$h(z)+gz=h_{0}$

$\int {\frac {dh}{dz}}dz=-\int gdz$

$h(z)=-gz+constant$

$[M]\{{\ddot {v}}\}+[K]\{v\}=\{F\}$

$\{v\}=[A]\{F\}$

$\omega ^{2}=13.424{\frac {EI}{\rho Al^{4}}}$

$[A]={\begin{bmatrix}1/3&5/6&4/3\\5/6&8/3&14/3\\4/3&14/3&9\end{bmatrix}}{\frac {1}{EI}}\left({\frac {l}{3}}\right)^{3}$

$\omega _{n}=k_{i}^{2}{\sqrt {\frac {EI}{\rho A}}}$

$cos(k_{i}l)cosh(k_{i}l)=-1$

$h=C_{p}T$

$c_{p}{\frac {dT}{dz}}=-g$

${\frac {dT}{dz}}={\frac {-g}{c_{p}}}$

$-\int {\frac {dT}{dz}}dz=\int {\frac {g}{c_{p}}}dz$

$-T(z)={\frac {g}{c_{p}}}z-constant$

$c_{p}constant=konstant=T(z)c_{p}+gz$

$konstant=h(z)+gz$

$-{\frac {dT}{dz}}={\frac {g}{c_{p}}}$

$T(z)=\psi z+T_{0}$

${\frac {dp}{\rho ^{2}}}$

$z_{dew}={\frac {1}{\psi }}\left({\frac {p_{SAT}(z_{dew})}{\beta }}-T_{0}\right)$