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Blowing in the wind

In this post, I look at the drag force in air. I make these assumptions:

• the quadratic relationship is a reasonable approximation, ie assume $k_1 =0$
• the issues of density change related to the relationship between altitude and temperature is ignored
• the body is spherical and the coefficient is 0.87 $r^2$ where r is the radius of the sphere. For simplicity we will take $r =1$.
The following graphic shows the change of speed with time in free fall without atmosphere and the fall with air drag force. Note the terminal velocity and the time to approach the asymptote.

The following graphic is distance against time:Finally, if the moon had the same gravitational acceleration as the earth, the falls on the moon and on earth of the identical spheres (equal masses) are simulated. Note the terminal velocity.

Notes:

• The above follow from solving:  $\frac{dv}{dt} =g- \frac{c_2r^2v^2}{m}$ using initial condition $v(0) =0$
• The terminal velocity: $v_\text{term} = \sqrt{\frac{mg}{c_2r^2}}$
• For spherical objects in air  $c_1\approx 3.1 \times 10^{-4}$ $\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}$.  So the contribution of the two coefficients  can be gauged using critical speed  (the speed at which the two drag terms contribute equally) $v_\text{crit} = c_1/(c_2 r)\approx 4 \times 10^-4/r$. So for our 1 m object in air, the quadratic term dominates and our assumption is justified. For our spherical objects of radius $r$, in the viscous region $v_\text{term} \propto r^2$ and in the pressure dominated region $v_\text{term} \propto r^{0.5}$
See Professor Lewin’s lectures on Resistive Forces.