Fluid Slows the Fall

A projectile of mass $m$ is launched from the point $(x,y) = (0,h)$ with speed $v_0$ purely in the horizontal $(x)$ direction. The projectile is subjected to a fluid drag force of the following form:

$\large{\vec{F}_{drag} = -\alpha \, |\vec{v}|^2 \, \hat{v}}$

In the above expression, $\vec{v}$ is the projectile's vector velocity, $\hat{v}$ is a unit-vector in the direction of the velocity, and $\alpha$ is a constant. Ambient gravity $g$ is in the negative $y$ direction.

If the time taken to fall to $y = 0$ without fluid drag is $t_0$ , and the time taken to fall to $y = 0$ with fluid drag is $t_f$ , determine the ratio $\large{\frac{t_f}{t_0}}$ .

Inspiration

Details and Assumptions:
- $m = 1 \, \text{kg}$
- $h = 10 \, \text{m}$
- $v_0 = 100 \, \text{m/s}$
- $\alpha = 0.05 \, \text{kg/m}$
- $g = 10 \, \text{m/s}^2$