Efecto de la resistencia del aire en la caída de los cuerpos

Al ser la densidad del aire pequeña, podemos despreciar el empuje frente al peso del cuerpo. La ecuación del movimiento de un cuerpo esférico de radio R y masa m es

$m \frac{d v}{d t} = m g - 0.2 ρ_{f} π R^{2} v^{2}$

Se ha tomado la dirección positiva hacia abajo

A medida que el cuerpo cae, se incrementa la velocidad, la fuerza de rozamiento crece hasta que se iguala al peso. El cuerpo se mueve con velocidad constante denominada velocidad límite v_T

$v_{T}^{2} = \frac{m g}{0.2 ρ_{f} π R^{2}}$

La ecuación del movimiento se escribe

$\frac{d v}{d t} = g (1 - \frac{v^{2}}{v_{T}^{2}})$

Integramos la ecuación diferencial con la condición inicial siguiente: en el instante t=0, la velocidad del cuerpo es v₀.

$\begin{array}{l} \int_{v_{0}}^{v} \frac{d v}{1 - v^{2} / v_{T}^{2}} = g \int_{0}^{t} d t \\ \int_{v_{0}}^{v} (\frac{\frac{1}{2}}{1 - v / v_{T}} + \frac{\frac{1}{2}}{1 + v / v_{T}}) d v = g t \\ {\frac{v_{T}}{2} \ln \frac{1 + v / v_{T}}{1 - v / v_{T}} |}_{v_{0}}^{v} = g t \\ \frac{1 + v / v_{T}}{1 - v / v_{T}} = \frac{1 + v_{0} / v_{T}}{1 - v_{0} / v_{T}} \exp (2 g t / v_{T}) \\ \frac{1 + v / v_{T}}{1 - v / v_{T}} = \exp (2 g t / v_{T} + 2 α) \\ v = v_{T} \frac{\exp (g t / v_{T} + α) - \exp (- g t / v_{T} - α)}{\exp (g t / v_{T} + α) + \exp (- g t / v_{T} - α)} \\ v = v_{T} \tanh (g t / v_{T} + α) \\ \exp (2 α) = \frac{1 + v_{0} / v_{T}}{1 - v_{0} / v_{T}} \\ \frac{v_{0}}{v_{T}} = \tanh α \\ v = v_{T} \tanh ((\frac{g t}{v_{T}}) + \tanh^{- 1} (\frac{v_{0}}{v_{T}})) \end{array}$

Integramos de nuevo para calcular la posición del cuerpo que cae en función del tiempo, sabiendo que en el instante t=0, parte del origen x=0.

$\begin{array}{l} x = \int_{0}^{t} v \cdot d t = \int_{0}^{t} v_{T} \tanh ((\frac{g t}{v_{T}}) + \tanh^{- 1} (\frac{v_{0}}{v_{T}})) \cdot d t = \\ \int \tanh u \cdot d u = \ln \cosh u + C \\ x = \frac{v_{T}^{2}}{g} \ln \frac{\cosh ((\frac{g t}{v_{T}}) + \tanh^{- 1} (\frac{v_{0}}{v_{T}}))}{\cosh (\tanh^{- 1} (\frac{v_{0}}{v_{T}}))} \\ \frac{\cosh (u + α)}{\cosh α} = \frac{\exp (u + α) - \exp (- u - α)}{\exp (α) - \exp (- α)} = \frac{\exp (u) \cdot \exp (2 α) - \exp (- u) + \exp (u) - \exp (- u) \cdot \exp (- 2 α)}{\exp (2 α) - \exp (- 2 α)} = \\ = \frac{\exp (u) \frac{1 + v_{0} / v_{T}}{1 - v_{0} / v_{T}} - \exp (- u) + \exp (u) - \exp (- u) \frac{1 - v_{0} / v_{T}}{1 + v_{0} / v_{T}}}{\frac{1 + v_{0} / v_{T}}{1 - v_{0} / v_{T}} - \frac{1 v_{0} / v_{T}}{1 + v_{0} / v_{T}}} = \frac{(\exp (u) - \exp (- u)) + \frac{v_{0}}{v_{T}} (\exp (u) + \exp (- u))}{2 \frac{v_{0}}{v_{T}}} = \\ \frac{v_{0}}{v_{T}} \sinh (u) + \cosh (u) \\ x = \frac{v_{T}^{2}}{g} \ln (\frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) + \cosh (\frac{g t}{v_{T}})) \end{array}$

Aproximaciones

Desarrollo en serie de la función x(t)

$\begin{array}{l} x (t) \approx x (0) + {\frac{d x}{d t} |}_{t = 0} t + \frac{1}{2!} {\frac{d^{2} x}{d t^{2}} |}_{t = 0} t^{2} + \frac{1}{3!} {\frac{d^{3} x}{d t^{3}} |}_{t = 0} t^{3} + \frac{1}{4!} {\frac{d^{4} x}{d t^{4}} |}_{t = 0} t^{4} + ... \\ x = \frac{v_{T}^{2}}{g} \ln (\frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) + \cosh (\frac{g t}{v_{T}})) \\ \frac{d x}{d t} = \frac{v_{T}^{2}}{g} \frac{\frac{g v_{0}}{v_{T}^{2}} \cosh (\frac{g t}{v_{T}}) + \frac{g}{v_{T}} \sinh (\frac{g t}{v_{T}})}{\frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) + \cosh (\frac{g t}{v_{T}})} = v_{T} \frac{\frac{v_{0}}{v_{T}} \cosh (\frac{g t}{v_{T}}) + \sinh (\frac{g t}{v_{T}})}{\frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) + \cosh (\frac{g t}{v_{T}})} \\ \frac{d^{2} x}{d t^{2}} = \frac{g (1 - \frac{v_{0}^{2}}{v_{T}^{2}})}{{(\frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) + \cosh (\frac{g t}{v_{T}}))}^{2}} \\ \frac{d^{3} x}{d t^{3}} = - \frac{2 g^{2}}{v_{T}} (1 - \frac{v_{0}^{2}}{v_{T}^{2}}) \frac{\frac{v_{0}}{v_{T}} \cosh (\frac{g t}{v_{T}}) + \sinh (\frac{g t}{v_{T}})}{{(\frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) + \cosh (\frac{g t}{v_{T}}))}^{3}} \\ \frac{d^{4} x}{d t^{4}} = - \frac{2 g^{3}}{v_{T}^{2}} (1 - \frac{v_{0}^{2}}{v_{T}^{2}}) \frac{1 - 2 \sinh^{2} (\frac{g t}{v_{T}}) - 4 \frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) \cosh (\frac{g t}{v_{T}}) - \frac{v_{0}^{2}}{v_{T}^{2}} (1 + 2 \cosh^{2} (\frac{g t}{v_{T}}))}{{(\frac{v_{0}}{v_{T}} \sinh (\frac{g t}{v_{T}}) + \cosh (\frac{g t}{v_{T}}))}^{4}} \\ x (t) \approx v_{0} t + \frac{1}{2} g (1 - \frac{v_{0}^{2}}{v_{T}^{2}}) t^{2} - \frac{g^{2} v_{0}}{3 v_{T}^{2}} (1 - \frac{v_{0}^{2}}{v_{T}^{2}}) t^{3} - \frac{g^{3}}{12 v_{T}^{2}} (1 + 3 \frac{v_{0}^{4}}{v_{T}^{4}} - 4 \frac{v_{0}^{2}}{v_{T}^{2}}) t^{4} + ... \end{array}$

Cuando el cuerpo se deja caer, v₀=0, parte del reposo

$x (t) \approx \frac{1}{2} g t^{2} - \frac{g^{3}}{12 v_{T}^{2}} t^{4} + ...$

Referencias

Lindemuth J. The effect o fair resistance on falling balls. Am. J. Phys. 39, July 1971, pp. 757-759