[Physics] the terminal velocity for a mobile phone

airgravityvelocity

You may have seen the story of the iPhone which was dropped from perhaps 13,500 feet by a skydiver – it survived.

This made me wonder how to work out the terminal velocity for something like that. Obviously calculating terminal velocity for a sphere can be relatively straightforward, but with a flattened oblong, what factors come into play?

End on will be fast, flat will be slow, but is there a stable configuration?

Best Answer

From the equilibrium between drag and weight: $$ \frac{1}{2} C_x \rho v^2 S = m g $$ we can write the terminal velocity as $$ v = \sqrt{\frac{2 m g}{C_x \rho S}} $$ where $m$ is the mass of the phone, $C_x$ its drag coefficient, $S$ its section, $g$ the acceleration of gravity, and $\rho$ the density of air.

Now, this is not really a good answer, as the big question is how to estimate $S$ (depends on the phone orientation) and $C_x$. But at least you can compute an order of magnitude, as in most cases $C_x$ of of order 1.