Free Fall Equation (Air Drag) (Q3836): Difference between revisions

Latest revision as of 10:11, 10 October 2024

Modeling the fall of objects by the laws of classical mechanics, including the aerodynamic drag and assuming a uniform gravitational field. Moreover, assuming the falling object to be a point mass.

Language	Label	Description	Also known as
English	Free Fall Equation (Air Drag)	Modeling the fall of objects by the laws of classical mechanics, including the aerodynamic drag and assuming a uniform gravitational field. Moreover, assuming the falling object to be a point mass.

Statements

instance of

Mathematical Formulation

0 references

defining formula

{\begin{aligned}m{\dot {v}}&=mg-{\frac {1}{2}}\rho C_{D}Av^{2}\\y(t)&=y_{0}+v_{0}t-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)\\v(t)&=v_{\infty }\tanh \left({\frac {gt}{v_{\infty }}}\right)\end{aligned}}

has characteristic

nonlinearity

1 reference

described at URL

https://en.wikipedia.org/wiki/Free_fall

in defining formula

v

symbol represents

Free Fall Velocity

0 references

m

symbol represents

Free Fall Mass

0 references

g

symbol represents

Gravitational Acceleration

0 references

\rho

symbol represents

Density of Air

0 references

C_{D}

symbol represents

Drag Coefficient

0 references

A

symbol represents

Cross Section

0 references

v_{0}

symbol represents

Free Fall Initial Velocity

0 references

v_{\infty }

symbol represents

Free Fall Terminal Velocity

0 references

t

symbol represents

Time

0 references

y

symbol represents

Free Fall Height

0 references

y_{0}

symbol represents

Free Fall Initial Height

0 references

community

MathModDB

0 references

Generalizes Formulation

Free Fall Equation (Vacuum)

Mathematical Condition

Vanishing Air Density

Vanishing Drag Coefficient

0 references

Identifiers

Wikidata QID

Q38083707

0 references

@@ Property / defining formula / Property / defining formula @@
- $m{\dot {v}}=mg-{\frac {1}{2}}\rho C_{D}Av^{2}$ 
m\dot{v}=mg-\frac{1}{2}\rho C_DAv^2
+ ${\begin{aligned}m{\dot {v}}&=mg-{\frac {1}{2}}\rho C_{D}Av^{2}\\y(t)&=y_{0}+v_{0}t-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)\\v(t)&=v_{\infty }\tanh \left({\frac {gt}{v_{\infty }}}\right)\end{aligned}}$ 
\begin{align} m\dot{v}&=mg-\frac{1}{2}\rho C_DAv^2 \\ y(t)&=y_0+v_0t-\frac{v_\infty^2}{g}\ln\cosh\left(\frac{gt}{v_\infty}\right)\\ v(t) &= v_{\infty}\tanh\left(\frac{gt}{v_{\infty}}\right) \end{align}
+has characteristic: nonlinearity
+described at URL: https://en.wikipedia.org/wiki/Free_fall
@@ Property / defining formula @@
- $y(t)=y_{0}+v_{0}t-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)$ 
y(t)=y_0+v_0t-\frac{v_\infty^2}{g}\ln\cosh\left(\frac{gt}{v_\infty}\right)
-Normal rank
@@ Property / Generalizes Formulation @@
-Free Fall Equation (Vacuum)
@@ Property / Generalizes Formulation: Free Fall Equation (Vacuum) / rank @@
-Normal rank
@@ Property / Generalizes Formulation: Free Fall Equation (Vacuum) / qualifier @@
-Vanishing Air Density:  $\rho \rightarrow 0$ 
\rho \rightarrow 0
@@ Property / Generalizes Formulation: Free Fall Equation (Vacuum) / qualifier @@
-Vanishing Drag Coefficient:  $C_{D}\rightarrow 0$ 
C_D \rightarrow 0
@@ Property / Generalizes Formulation @@
+Free Fall Equation (Vacuum)
@@ Property / Generalizes Formulation: Free Fall Equation (Vacuum) / rank @@
+Normal rank
@@ Property / Generalizes Formulation: Free Fall Equation (Vacuum) / qualifier @@
+Mathematical Condition: Vanishing Air Density
@@ Property / Generalizes Formulation: Free Fall Equation (Vacuum) / qualifier @@
+Mathematical Condition: Vanishing Drag Coefficient

Free Fall Equation (Air Drag) (Q3836): Difference between revisions

$m{\dot {v}}=mg-{\frac {1}{2}}\rho C_{D}Av^{2}$

${\begin{aligned}m{\dot {v}}&=mg-{\frac {1}{2}}\rho C_{D}Av^{2}\\y(t)&=y_{0}+v_{0}t-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)\\v(t)&=v_{\infty }\tanh \left({\frac {gt}{v_{\infty }}}\right)\end{aligned}}$

$y(t)=y_{0}+v_{0}t-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)$

$\rho \rightarrow 0$

$C_{D}\rightarrow 0$

Latest revision as of 10:11, 10 October 2024

Statements

Identifiers

Sitelinks

mathematics(0 entries)

Free Fall Equation (Air Drag) (Q3836): Difference between revisions

m v ˙ = m g − 1 2 ρ C D A v 2 {\displaystyle m{\dot {v}}=mg-{\frac {1}{2}}\rho C_{D}Av^{2}}

y ( t ) = y 0 + v 0 t − v ∞ 2 g ln ⁡ cosh ⁡ ( g t v ∞ ) {\displaystyle y(t)=y_{0}+v_{0}t-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)}

ρ → 0 {\displaystyle \rho \rightarrow 0}

C D → 0 {\displaystyle C_{D}\rightarrow 0}

Latest revision as of 10:11, 10 October 2024

Statements

Identifiers

Sitelinks

mathematics(0 entries)

$m{\dot {v}}=mg-{\frac {1}{2}}\rho C_{D}Av^{2}$

$y(t)=y_{0}+v_{0}t-{\frac {v_{\infty }^{2}}{g}}\ln \cosh \left({\frac {gt}{v_{\infty }}}\right)$

$\rho \rightarrow 0$

$C_{D}\rightarrow 0$