Physics 303
One-Dimensional Free Fall

1. In solving Newton's Second Law for free fall through a resistive medium we used the following expression to represent the drag force
   

   $$\vec F_f = - \frac{1}{2} C_D \rho S v^2 \hat v$$ (1)

   
   
   where $C_D$ is the drag coefficient, $\rho$ is the density, $S$ is the cross-sectional area, $v$ is velocity, and $\hat v$ is a unit vector in the direction of $\vec v$. We found that $v$ and $t$ are related by
   

   $$\frac{1}{2 v_t} \ln \left ( \frac{v_t + v}{v_t - v} \right ) = -\frac{g}{v_t^2}t$$ (2)

   
   
   where
   

   $$v_t = \frac{2mg}{C_D\rho S}$$ (3)

   
   
   and $m$ is the mass and $g$ is the acceleration of gravity. Show that Equation 2 can be written in the following form.
   

   $$v(t) = - v_t \left [ \frac{1 - \exp(-\frac{2gt}{v_t})}{1 + \exp(-\frac{2gt}{v_t})} \right ]$$ (4)

   
   

2. Show that Equation 4 can be written in the following form
   

   $$v(t) = - v_t \tanh \left ( \frac{gt}{v_t} \right )$$ (5)

   
   
   where $\tanh x$ means the hyperbolic tangent of $x$ defined in the following way.
   

   $$\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$ (6)

   
   

3. Apply a direct integration to Equation 5 to get the general solution for $y(t)$ and then use the initial conditions that at $t=0$, $y=y_0$ to find the particular solution.

4. Use the solution from Problem 3 to show that time it takes for Ms. Vulovic to hit the ground can be written in the following way
   

   $$t_{hit} = \frac{v_t}{g} {\rm arccosh} \left [ \exp \left ( \frac{g y_0}{v_t^2} \right ) \right ]$$ (7)

   
   
   where ${\rm arccosh} x$ means the inverse hyperbolic cosine of $x$.